#125350

order	512
nilpotency class	4
isoclinism	isoclinic class of #125287
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
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 ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,e^c=ei,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, 32546);