#125538

order	512
nilpotency class	4
isoclinism	isoclinic class of #125537
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

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

normal subgroups only

This group has 306 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fg,d^2=gi,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=efg,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, 32734);