#124301

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fi,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=dg,e^b=efg,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31497);