#7312307

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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

1681 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7219503);