#6537623

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 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)(20,33,35,40)(21,26,23,36)(25,37,41,34)(29,44,45,48)(31,43,32,39)(38,46,42,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,38)(26,39)(27,28)(33,48)(34,46)(35,45)(36,43)(37,47)(40,44)(41,42);(10,13)(14,16)(17,20,22,35)(18,25,27,41)(19,29,30,45)(21,34,23,37)(24,38,28,42)(26,40,36,33)(31,46,32,47)(39,44,43,48);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,28)(29,45)(31,32)(38,42)(39,43)(44,48)(46,47);(11,15)(14,16)(17,21,22,23)(18,26,27,36)(19,31,30,32)(20,34,35,37)(24,39,28,43)(25,40,41,33)(29,46,45,47)(38,44,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,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, 6444819);