#125422

order	512
nilpotency class	4
isoclinism	isoclinic class of #125422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 66 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)(4,6)(7,8)(9,10,13,17,16,21,12,19,14,22,31,35,15,20,30,34)(11,23,26,36,24,18,25,38,27,39,32,37,28,40,29,33);(2,5)(3,8)(6,7)(10,17)(11,24)(12,13)(15,16)(18,36)(19,20)(21,34)(22,35)(23,38)(26,32)(27,28)(30,31)(33,39)(37,40);(2,6)(5,7)(9,11,16,24,14,27,15,28)(10,18,20,23,22,40,21,39)(12,29,31,26,30,25,13,32)(17,33,34,37,35,38,19,36)

Polycyclic presentation:

2^9|a^2=e,c^2=fg,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,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^c=ei,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, 32618);