#123168

order	512
nilpotency class	4
isoclinism	isoclinic class of #123151
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂³	#447
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

254 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=fh,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30364);