#7307038

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^c=ei,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, 7214234);