#122512

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	C₂×#135	C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

275 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=fi,e^2=g,f^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,d^b=dfhi,e^b=ehi,f^b=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29708);