#330772

order	512
nilpotency class	3
isoclinism	isoclinic class of #329395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=eh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237968);