#233616

order	512
nilpotency class	3
isoclinism	isoclinic class of #232749
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140812);