#7086995

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1331 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)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,27,30,42)(20,24,32,41)(21,34,36,26)(23,35,38,29)(28,43,31,45)(33,48,47,40)(37,46,39,44);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,35)(22,30)(24,39)(25,38)(26,42)(27,34)(28,32)(29,36)(33,44)(37,41)(40,43)(45,48)(46,47);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,43)(29,44)(30,31)(34,48)(35,46)(36,47)(38,39)(42,45);(11,16)(13,15)(19,29)(21,36)(23,27)(26,34)(28,44)(30,35)(31,46)(33,47)(37,43)(38,42)(39,45)(40,48);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(18,25)(19,30)(20,32)(26,34)(29,35)(33,47)(37,39)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,e^2=g,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, 6994191);