#6608969

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

1447 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)(19,24)(20,32)(21,26)(22,36)(23,27)(25,35)(28,41)(29,38)(30,39)(31,45)(33,46)(34,40)(37,43)(42,47)(44,48);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,44)(34,42)(35,43)(36,45)(40,47)(46,48);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(17,20,22,35,23,33,21,34)(18,25,26,32,27,40,36,46)(19,28,31,43,29,44,30,42)(24,37,39,41,38,47,45,48);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(19,29)(20,33)(24,38)(25,40)(30,31)(32,46)(34,35)(39,45);(11,12)(14,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,34,33,35)(24,39,38,45)(25,32,40,46)(28,42,44,43)(37,41,47,48)

Polycyclic presentation:

2^9|b^2=h,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh

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, 6516165);