#96985

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#555	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

724 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4181);