#500108

order	512
nilpotency class	4
isoclinism	isoclinic class of #499904
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3192
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24005
prime radical	#60453	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

977 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 977 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,8)(4,17)(5,21)(6,26)(7,11)(9,19)(10,25)(12,24)(13,38)(14,29)(15,44)(16,47)(18,50)(20,35)(22,34)(23,54)(27,51)(28,40)(30,46)(31,43)(32,60)(33,52)(36,55)(37,53)(39,49)(41,58)(42,63)(45,56)(48,61)(57,64)(59,62);(1,3)(2,8)(4,13)(5,15)(6,16)(7,14)(9,32)(10,30)(11,29)(12,31)(17,49)(18,39)(19,40)(20,41)(21,44)(22,45)(23,42)(24,43)(25,46)(26,47)(27,48)(28,33)(34,59)(35,61)(36,57)(37,56)(38,50)(51,58)(52,60)(53,62)(54,63)(55,64);(1,4,6,27,7,18,24,20)(2,9,12,37,11,33,26,34)(3,13,16,48,14,39,43,41)(5,22,25,52,23,53,55,19)(8,28,31,59,29,32,47,56)(10,35,21,50,36,51,54,17)(15,45,46,60,42,62,64,40)(30,58,44,49,57,61,63,38);(3,14)(5,23)(8,29)(10,36)(13,39)(16,43)(17,50)(19,52)(21,54)(22,53)(25,55)(28,32)(31,47)(35,51)(41,48)(56,59)

Polycyclic presentation:

2^9|c^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^c=ei,f^c=fhi,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307945, #72307958, #72308066
maximal normal subgroup of	#72307945, #72307958, #72308066

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407304);