#125195

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
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 ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 332 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=hi,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,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, 32391);