#125113

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

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

308 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gi,d^2=gi,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=efg,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, 32309);