#6609063

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,22,26)(19,28,30,23)(20,32,35,42)(21,25,36,27)(24,40,43,34)(29,44,33,47)(31,39,46,38)(37,45,41,48);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,23,26,28)(20,33,35,29)(21,31,36,46)(24,41,43,37)(25,38,27,39)(32,44,42,47)(34,48,40,45);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(17,20,21,34,22,35,36,40)(18,24,27,32,26,43,25,42)(19,29,31,45,30,33,46,48)(23,37,39,47,28,41,38,44);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(19,30)(23,28)(29,33)(31,46)(37,41)(38,39)(44,47)(45,48);(11,12)(14,16)(17,21,22,36)(18,25,26,27)(19,31,30,46)(20,34,35,40)(23,38,28,39)(24,42,43,32)(29,45,33,48)(37,44,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,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, 6516259);