#6516489

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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

1387 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi

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, 6423685);