#6551923

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(3,8)(4,6)(5,7)(9,10,15,14)(11,12,13,16)(17,18,22,28,23,26,21,27)(19,24,32,39,30,40,31,38)(20,33,34,41,35,42,36,25)(29,43,45,47,44,48,46,37);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,16,14,12)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,37)(26,40)(27,38)(28,39)(33,43)(34,45)(35,44)(36,46)(41,47)(42,48);(1,4,5,3)(2,6,7,8)(9,12,15,16)(10,11,14,13)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,41)(27,33)(28,42)(30,44)(31,45)(32,46)(38,43)(39,48)(40,47);(9,13,15,11)(10,16,14,12)(18,26)(19,30)(25,41)(27,28)(29,44)(31,32)(33,42)(45,46);(9,14,15,10)(11,12,13,16)(17,21,23,22)(18,27,26,28)(19,31,30,32)(20,34,35,36)(24,38,40,39)(25,33,41,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6459119);