#7487209

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1314 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7394405);