#6623015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 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)(3,7)(4,6)(5,8)(10,14)(15,16)(17,18)(19,28)(20,32)(21,26)(22,27)(23,38)(24,31)(25,36)(29,44)(30,41)(33,47)(34,46)(35,43)(37,40)(39,48)(42,45);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,24)(20,33)(21,37)(22,31)(23,30)(25,42)(26,41)(27,28)(29,34)(32,44)(35,45)(36,48)(38,40)(39,43)(46,47);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,21,35,22,34,23,36)(18,25,38,46,27,43,26,32)(19,29,37,48,31,33,30,45)(24,39,40,44,28,42,41,47);(1,4,5,3)(2,6,8,7)(11,12)(15,16)(20,34)(25,43)(29,33)(32,46)(35,36)(39,42)(44,47)(45,48);(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,26,27,38)(19,30,31,37)(20,35,34,36)(24,40,28,41)(25,32,43,46)(29,45,33,48)(39,44,42,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=fg,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6530211);