#6466355

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373551);