#509848

order	512
nilpotency class	4
isoclinism	isoclinic class of #509117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3199
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#31056
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

798 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=ef,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=eh,f^d=fi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417044);