#126515

order	512
nilpotency class	4
isoclinism	isoclinic class of #125651
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1655
commutator, Frattini, cosocle	#376	C₂³
socle	C₂	#7833
prime radical	#532	D₄
LCS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #376 ◁ G
LECS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

249 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,d^2=f,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=df,e^b=egi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33711);