#125716

order	512
nilpotency class	4
isoclinism	isoclinic class of #125636
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1954 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

458 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,d^b=df,e^b=eg,f^b=fi,d^c=dg,e^c=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32912);