#6431777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431743
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#392
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6969 conjugacy classes of subgroups; only the 984 normal subgroups are shown.

normal subgroups only

This group has 6969 conjugacy classes of subgroups — too many to draw. Only its 984 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,14)(4,20)(5,11)(6,29)(7,12)(8,13)(9,17)(10,24)(15,42)(16,41)(18,33)(19,32)(21,40)(22,49)(23,37)(25,36)(26,38)(27,54)(28,39)(30,44)(31,46)(34,48)(35,53)(43,62)(45,57)(47,56)(50,61)(51,60)(52,59)(55,63)(58,64);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,34)(11,31)(12,32)(13,33)(14,29)(15,23)(20,30)(22,50)(24,44)(25,48)(26,45)(27,47)(28,46)(35,58)(36,42)(37,40)(38,56)(39,57)(41,54)(43,52)(49,55)(51,64)(53,63)(59,61)(60,62);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,30)(11,35)(12,36)(13,37)(14,40)(16,43)(17,21)(18,48)(19,44)(20,29)(26,51)(27,52)(28,53)(31,55)(32,34)(33,42)(38,59)(39,60)(41,61)(45,63)(46,50)(47,64)(49,54)(56,62)(57,58);(1,5)(2,11)(3,16)(4,22)(6,27)(7,28)(8,26)(9,31)(10,35)(12,39)(13,38)(14,41)(15,43)(17,46)(18,47)(19,45)(20,49)(21,50)(23,52)(24,53)(25,51)(29,54)(30,55)(32,57)(33,56)(34,58)(36,60)(37,59)(40,61)(42,62)(44,63)(48,64);(5,26)(11,38)(16,45)(22,51)(27,28)(31,56)(35,59)(39,54)(41,57)(43,63)(46,47)(49,60)(50,64)(52,53)(55,62)(58,61)

Polycyclic presentation:

2^9|f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,f^b=fi,h^b=hi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285753, #72286055, C₂×#6431777
maximal normal subgroup of	#72285753, #72286055, C₂×#6431777
maximal quotient of	C₂×#6431777

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338973);