#125353

order	512
nilpotency class	4
isoclinism	isoclinic class of #125094
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

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

normal subgroups only

This group has 302 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) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32549);