#125112

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=g,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, 32308);