#7083411

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 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)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,27,30,43)(20,24,32,40)(21,34,35,26)(23,36,38,29)(28,42,31,47)(33,48,44,41)(37,45,39,46);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,29,22,30,35,36)(18,23,26,27,25,38,34,43)(20,31,33,45,32,28,44,46)(24,39,41,47,40,37,48,42);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,40)(26,41)(27,42)(29,46)(30,31)(34,48)(35,44)(36,45)(38,39)(43,47);(1,4,5,3)(2,7,8,6)(11,16)(13,15)(19,29)(20,32)(21,35)(23,27)(24,40)(26,34)(28,45)(30,36)(31,46)(37,47)(38,43)(39,42);(10,14)(13,15)(18,25)(19,30)(20,32)(26,34)(29,36)(33,44)(37,39)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,d^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,d^c=dh,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, 6990607);