#123190

order	512
nilpotency class	4
isoclinism	isoclinic class of #123171
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

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

normal subgroups only

This group has 264 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)(7,13)(10,15)(14,16)(17,18,24,30)(19,31,20,34)(21,37,41,46)(22,29,23,28)(25,35,44,32)(26,33,45,36)(27,39,42,40)(38,47,48,43);(7,12)(9,15)(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)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(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|a^2=h,b^2=h,c^2=ghi,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=eh,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30386);