#500852

order	512
nilpotency class	4
isoclinism	isoclinic class of #500850
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3058
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24267
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

934 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 160 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)(3,17)(4,24)(5,28)(6,32)(7,16)(8,14)(9,15)(10,21)(11,26)(12,29)(13,18)(19,54)(20,56)(22,58)(23,50)(25,60)(27,47)(30,62)(31,48)(33,45)(34,44)(35,49)(36,42)(37,53)(38,61)(39,64)(40,57)(41,63)(43,55)(46,51)(52,59);(1,3,18,51,8,22,34,57,9,23,33,64,7,35,6,21)(2,10,32,49,16,39,45,50,15,40,44,58,14,46,13,17)(4,19,52,63,26,30,61,29,27,5,20,53,36,31,25,55)(11,41,59,54,24,43,60,48,42,37,56,28,47,12,38,62);(1,4)(2,11)(3,19)(5,23)(6,25)(7,36)(8,26)(9,27)(10,37)(12,39)(13,38)(14,47)(15,42)(16,24)(17,48)(18,52)(20,33)(21,55)(22,30)(28,49)(29,57)(31,35)(32,59)(34,61)(40,41)(43,46)(44,56)(45,60)(50,62)(51,63)(53,64)(54,58);(1,5,18,53,8,31,34,55,9,19,33,63,7,30,6,29)(2,12,32,62,16,41,45,54,15,43,44,48,14,37,13,28)(3,20,51,36,22,25,57,4,23,52,64,26,35,61,21,27)(10,38,49,11,39,59,50,24,40,60,58,42,46,56,17,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217290, #72224996, #72225016, …
maximal normal subgroup of	#72217290, #72224996, #72225016, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408048);