#7283634

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
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	#24455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7190830);