#122330

order	512
nilpotency class	4
isoclinism	isoclinic class of #11692
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×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂²	C₂×#324
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

278 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,6,11,3,8)(4,7,12,15,13,9,14,16)(17,18,21,26,22,27,36,44,23,28,37,45,20,25,34,42)(19,29,31,46,30,40,47,43,32,41,35,48,33,24,38,39);(2,7,11,9)(3,5)(8,16,10,15)(12,14)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,41)(26,43)(27,29)(28,40)(31,37)(34,47)(36,38)(39,45)(42,46)(44,48);(1,3,6,5)(2,8,11,10)(4,12,13,14)(7,15,9,16)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=hi,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=di,e^b=efhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29526);