#6549196

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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

1383 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)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,25,36,42)(19,27,31,38,30,23,29,39)(20,32,35,40,34,41,33,24)(28,44,45,37,46,48,47,43);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,22,30)(18,23,25,27)(20,28,34,46)(21,31,36,29)(24,37,40,43)(26,39,42,38)(32,48,41,44)(33,47,35,45);(10,11)(14,15)(17,20)(18,24)(19,28)(21,33)(22,34)(23,37)(25,40)(26,41)(27,43)(29,45)(30,46)(31,47)(32,42)(35,36)(38,48)(39,44);(1,4,5,3)(2,6,7,8)(12,16)(14,15)(18,25)(23,27)(24,40)(26,42)(32,41)(37,43)(38,39)(44,48);(9,12,13,16)(10,14,11,15)(17,21,22,36)(18,26,25,42)(19,29,30,31)(20,33,34,35)(23,38,27,39)(24,41,40,32)(28,45,46,47)(37,48,43,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6456392);