#122603

order	512
nilpotency class	4
isoclinism	isoclinic class of #122554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

334 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=hi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

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, 29799);