#7090010

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,25,20,31)(21,26,36,44)(22,37,39,27)(24,33,41,30)(29,42,32,47)(34,40,46,48)(35,45,38,43);(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,28,41)(21,34,36,46)(22,33,39,30)(25,27,31,37)(26,42,44,47)(29,35,32,38)(40,43,48,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,19)(18,25,28,31)(21,32,36,29)(22,33,39,30)(24,27,41,37)(26,42,44,47)(34,38,46,35)(40,45,48,43);(11,16)(13,15)(17,21,23,36)(18,26,28,44)(19,29,20,32)(22,38,39,35)(24,40,41,48)(25,42,31,47)(27,45,37,43)(30,34,33,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,20)(24,41)(25,31)(29,32)(30,33)(34,46)(40,48)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,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, 6997206);