#6537683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444879);