#125248

order	512
nilpotency class	4
isoclinism	isoclinic class of #125076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

286 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fghi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32444);