#7085914

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1359 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=h,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=eh,d^c=dh,e^c=ei,e^d=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, 6993110);