#123202

order	512
nilpotency class	4
isoclinism	isoclinic class of #123191
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

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

normal subgroups only

This group has 242 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)(3,6)(4,7)(5,9)(8,13)(10,15)(11,16)(12,14)(17,18,22,27)(19,28,30,24)(20,32,35,40)(21,37,34,26)(23,39,42,29)(25,44,47,36)(31,43,38,45)(33,46,48,41);(1,3)(2,6)(4,12)(5,10)(7,16)(8,14)(9,15)(11,13)(17,19,22,30)(18,23,27,42)(20,33,35,48)(21,38,34,31)(24,32,28,40)(25,45,47,43)(26,46,37,41)(29,44,39,36);(2,7,9,8)(3,10)(6,14,15,16)(11,12)(17,19)(18,24,27,28)(20,29)(21,31)(22,30)(23,40,42,32)(25,46,47,41)(26,43,37,45)(33,36)(34,38)(35,39)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30398);