#125216

order	512
nilpotency class	4
isoclinism	isoclinic class of #125064
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

346 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,8,6,3,4,7)(9,10,13,17,14,19,30,18)(11,20,23,39,21,38,29,35)(12,24,26,16,28,36,27,37)(15,31,33,40,32,25,34,22);(1,3,6,2)(4,7,5,8)(9,11,14,21)(10,15)(12,25,28,31)(13,29,30,23)(16,35)(17,33)(18,34)(19,32)(20,36)(22,26,40,27)(24,38)(37,39);(1,4,6,5)(2,7,3,8)(11,21)(15,32)(20,38)(22,40)(23,29)(25,31)(33,34)(35,39)

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=hi,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efg,f^b=fi,g^b=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32412);