#99698

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#386
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

390 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

(2,6,7,8)(10,13,14,16)(17,18,21,25,23,26,22,31)(19,27,28,32,20,29,30,24);(1,2,4,8,5,7,3,6)(11,15)(13,16)(17,19,23,20)(18,24,26,32)(21,30,22,28)(25,29,31,27);(1,3,5,4)(2,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,20,23,19)(18,24,26,32)(21,30,22,28)(25,27,31,29)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=hi,f^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6894);