#126465

order	512
nilpotency class	4
isoclinism	isoclinic class of #125887
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1659
commutator, Frattini, cosocle	#376	C₂³
socle	C₂	#7821
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

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

normal subgroups only

This group has 237 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fgh,d^2=fi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,d^b=dfi,e^b=egi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,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, 33661);