#500853

order	512
nilpotency class	4
isoclinism	isoclinic class of #500851
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

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

normal subgroups only

This group has 930 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)(7,15)(8,16)(9,14)(10,21)(11,26)(12,29)(13,18)(19,54)(20,57)(22,60)(23,51)(25,38)(27,48)(30,61)(31,49)(33,47)(34,46)(35,50)(36,43)(37,53)(39,63)(40,59)(41,52)(42,58)(44,62)(45,56)(55,64);(1,3,18,52,9,22,34,59,8,23,33,63,7,35,6,21)(2,10,32,50,15,39,47,51,16,40,46,60,14,41,13,17)(4,5,20,53,26,31,58,56,27,19,55,62,36,30,25,29)(11,37,57,28,24,12,38,61,43,44,64,54,48,45,42,49);(1,4)(2,11)(3,19)(5,23)(6,25)(7,36)(8,27)(9,26)(10,37)(12,39)(13,42)(14,48)(15,24)(16,43)(17,49)(18,20)(21,56)(22,30)(28,50)(29,59)(31,35)(32,57)(33,55)(34,58)(38,47)(40,44)(41,45)(46,64)(51,61)(52,62)(53,63)(54,60);(1,5,18,53,9,31,34,56,8,19,33,62,7,30,6,29)(2,12,32,61,15,44,47,54,16,45,46,49,14,37,13,28)(3,20,52,26,22,58,59,27,23,55,63,36,35,25,21,4)(10,38,50,43,39,64,51,48,40,42,60,11,41,57,17,24)

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,d^c=di,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72224995, #72225015, …
maximal normal subgroup of	#72217287, #72224995, #72225015, …
maximal quotient of	C₂×#500853

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408049);