#125546

order	512
nilpotency class	4
isoclinism	isoclinic class of #125537
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

236 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,7)(6,8)(9,10,13,19,15,20,29,37,14,21,30,17,28,36,12,18)(11,22,24,16,26,32,23,33,25,35,27,34,31,40,39,38);(1,2,4,6)(3,8,7,5)(9,11,14,25)(10,16)(12,27,29,24)(13,23,30,39)(15,31,28,26)(17,35)(18,32)(19,22)(20,38)(21,34)(33,36)(37,40);(9,12,28,30,14,29,15,13)(10,17,20,18,21,19,36,37)(11,23,31,24,25,39,26,27)(16,32,33,35,34,40,38,22)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32742);