#122329

order	512
nilpotency class	4
isoclinism	isoclinic class of #11692
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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 ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 293 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)(3,9,10,15)(4,12,13,7)(5,8)(6,11)(14,16)(17,18,20,25,21,26,31,39,22,24,30,36,32,37,43,46)(19,23,28,34,29,35,42,45,27,33,40,38,41,44,48,47);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19)(18,23,24,33)(20,28)(21,29)(22,27)(25,38,36,34)(26,35,37,44)(30,40)(31,42)(32,41)(39,47,46,45)(43,48);(3,10)(6,14)(9,15)(11,16)(19,27)(23,33)(28,40)(29,41)(34,38)(35,44)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=e,b^2=f,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, 29525);