#7020310

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	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1903 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,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, 6927506);