#7195116

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(6,8)(10,13)(11,15)(17,18)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,39)(32,47)(33,41)(34,38)(35,48)(36,43)(37,44)(40,45)(42,46);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,22,30)(18,23,27,26)(20,28,34,33)(21,29,37,36)(24,38,41,31)(25,39,44,43)(32,45,35,46)(40,48,42,47);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,20,30,33,22,34,19,28)(18,24,23,38,27,41,26,31)(21,35,36,45,37,32,29,46)(25,42,39,47,44,40,43,48);(2,5)(6,8)(10,13)(14,16)(17,21,22,37)(18,25,27,44)(19,29,30,36)(20,32,34,35)(23,39,26,43)(24,40,41,42)(28,45,33,46)(31,47,38,48);(9,11,12,15)(10,14,13,16)(17,19,22,30)(18,23,27,26)(20,33,34,28)(21,36,37,29)(24,31,41,38)(25,43,44,39)(32,45,35,46)(40,48,42,47)

Polycyclic presentation:

2^9|b^2=i,c^2=fghi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7102312);