#295113

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202309);