#125536

order	512
nilpotency class	4
isoclinism	isoclinic class of #125534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 305 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)(9,10,13,17,14,19,28,36,15,20,29,37,27,18,12,26)(11,21,23,38,24,16,31,32,25,33,40,35,30,34,22,39);(2,5)(3,8)(6,7)(9,11,15,25)(10,16,20,34)(12,23,28,40)(13,22,29,31)(14,30,27,24)(17,35,37,38)(18,33,19,21)(26,39,36,32);(9,12,27,29,15,28,14,13)(10,17,19,36,20,37,18,26)(11,22,30,40,25,31,24,23)(16,32,33,35,34,39,21,38)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fgi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,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, 32732);