#122514

order	512
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	C₂²×D₈	C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 291 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) = 128

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

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

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,c^b=ch,d^b=dfhi,e^b=ehi,f^b=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29710);