#7312363

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

1683 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)(19,25,33,44,35,42,34,41)(20,36,37,26,21,27,38,43)(31,45,47,39,32,40,46,48);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,31)(21,32)(22,34)(23,35)(24,33)(26,39)(27,40)(28,44)(29,41)(30,42)(36,45)(37,47)(38,46)(43,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,26,30,43)(19,31,35,32)(22,37,24,38)(25,39,42,48)(27,28,36,29)(33,46,34,47)(40,44,45,41);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,23,20)(18,27,30,36)(19,32,35,31)(22,37,24,38)(25,40,42,45)(26,28,43,29)(33,46,34,47)(39,44,48,41);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,33,35,34)(20,37,21,38)(25,41,42,44)(26,36,43,27)(31,46,32,47)(39,40,48,45)

Polycyclic presentation:

2^9|a^2=gi,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=ehi,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, 7219559);