#125132

order	512
nilpotency class	4
isoclinism	isoclinic class of #125098
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×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

292 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,7)(6,8)(10,14,15,16)(17,18,20,26,22,28,31,25,23,29,30,32,21,27,19,24);(1,2,4,6)(3,8,7,5)(9,10)(11,14)(12,16)(13,15)(18,24)(19,20)(21,22)(25,29)(26,27)(28,32)(30,31);(9,11,13,12)(10,14,15,16)(17,19,21,30,23,31,22,20)(18,25,27,26,29,24,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32328);