#7090510

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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1668 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)(18,23)(19,28)(21,24)(25,38)(26,31)(27,36)(29,43)(33,34)(35,40)(39,47)(41,42)(44,48);(9,10)(11,15)(12,14)(13,16)(17,18,24,36,22,27,21,23)(19,25,37,38,28,40,30,35)(20,31,34,41,32,42,33,26)(29,44,46,48,43,47,45,39);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,35)(24,37)(26,39)(27,40)(31,44)(32,43)(33,45)(34,46)(36,38)(41,48)(42,47);(11,16)(13,15)(17,20,22,32)(18,26,27,41)(19,29,28,43)(21,34,24,33)(23,31,36,42)(25,39,40,48)(30,46,37,45)(35,44,38,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,28)(20,32)(25,40)(26,41)(30,37)(31,42)(33,34)(35,38)

Polycyclic presentation:

2^9|b^2=fi,c^2=ghi,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, 6997706);