#123100

order	512
nilpotency class	4
isoclinism	isoclinic class of #123091
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	#3245	C₂²
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

298 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 42 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)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18)(19,31)(20,34)(21,37)(22,29)(23,28)(24,30)(25,32)(26,36)(27,40)(33,45)(35,44)(38,47)(39,42)(41,46)(43,48);(7,12)(9,16)(17,19,24,20)(18,25,30,44)(21,38,41,48)(22,33,23,36)(26,42,45,27)(28,43,29,47)(31,46,34,37)(32,39,35,40);(1,3,5,10)(2,6,8,15)(4,13,14,11)(7,16,12,9)(17,20,24,19)(18,26,30,45)(21,35,41,32)(22,33,23,36)(25,42,44,27)(28,31,29,34)(37,43,46,47)(38,40,48,39)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=fh,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,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, 30296);