#7078782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1414 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,30,35,42)(20,26,38,25)(21,28,22,27)(24,33,43,34)(31,47,36,41)(32,40,48,44)(37,45,39,46);(9,10)(11,15)(12,14)(13,16)(17,19,23,35)(18,24,29,43)(20,36,38,31)(21,34,22,33)(25,44,26,40)(27,30,28,42)(32,37,48,39)(41,46,47,45);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,31)(21,39)(22,37)(23,38)(24,40)(26,29)(27,46)(28,45)(30,41)(32,33)(34,48)(35,36)(42,47)(43,44);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20)(18,26)(19,32)(21,37)(22,39)(23,38)(24,41)(25,29)(27,46)(28,45)(30,40)(31,33)(34,36)(35,48)(42,44)(43,47);(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,33,35,34)(20,37,38,39)(24,42,43,30)(25,45,26,46)(31,48,36,32)(40,41,44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6985978);