#7020176

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

1926 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)(19,24)(20,30)(21,26)(22,37)(23,27)(25,32)(28,41)(29,38)(31,47)(33,40)(34,48)(35,43)(36,44)(39,46)(42,45);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,36,35)(25,30,41,40)(26,38,44,43)(31,45,34,46)(39,47,42,48);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,32,23,33,19,28)(18,25,24,30,27,41,37,40)(21,34,35,45,36,31,29,46)(26,42,38,48,44,39,43,47);(3,8)(6,7)(11,16)(13,15)(17,21)(18,26)(19,29)(20,31)(22,35)(23,36)(24,38)(25,39)(27,44)(28,45)(30,47)(32,46)(33,34)(37,43)(40,48)(41,42);(10,14)(13,15)(17,19,23,22)(18,24,27,37)(20,32,33,28)(21,35,36,29)(25,40,41,30)(26,43,44,38)(31,45,34,46)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,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=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, 6927372);