#6608981

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgi,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, 6516177);