#6623027

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

1490 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)(20,33)(21,25)(22,26)(23,31)(24,35)(27,37)(29,44)(30,43)(32,39)(34,47)(36,41)(38,45)(40,46)(42,48);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,19)(18,23)(20,34)(21,32)(22,31)(24,40)(25,43)(26,28)(27,39)(29,36)(30,37)(33,44)(35,45)(38,42)(41,47)(46,48);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,21,35,22,36,37,48)(18,24,27,33,26,42,25,41)(19,29,32,46,31,34,30,45)(23,38,39,47,28,40,43,44);(1,4,5,3)(2,6,8,7)(11,12)(15,16);(9,12,13,11)(10,16,14,15)(17,21,22,37)(18,25,26,27)(19,30,31,32)(20,35,36,48)(23,39,28,43)(24,41,42,33)(29,45,34,46)(38,47,40,44)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,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=dhi,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6530223);