#7228807

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

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, 7136003);