#7078713

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1394 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)(17,18,22,26)(19,27,30,40)(20,31,32,24)(21,34,36,25)(23,35,38,29)(28,42,44,41)(33,39,47,46)(37,43,45,48);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,38)(20,28,32,44)(21,35,36,29)(24,37,31,45)(25,40,34,27)(33,48,47,43)(39,42,46,41);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,39)(26,31)(27,41)(29,43)(30,44)(34,46)(35,48)(36,47)(38,45)(40,42);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,29)(21,36)(23,27)(25,34)(28,43)(30,35)(33,47)(37,41)(38,40)(39,46)(42,45)(44,48);(10,14)(13,15)(19,30)(20,32)(23,38)(24,31)(27,40)(29,35)(33,47)(39,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^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, 6985909);