#6466409

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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

1311 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)(19,29)(20,25)(21,35)(22,27)(23,28)(24,31)(26,37)(30,39)(32,41)(33,42)(34,47)(36,40)(38,48)(43,45)(44,46);(2,5)(6,8)(17,19,22,31)(18,24,27,29)(20,33,32,23)(21,36,37,34)(25,28,41,42)(26,40,35,47)(30,45,46,38)(39,48,44,43);(9,11,12,16)(10,13,14,15)(17,20,23,19,22,32,33,31)(18,25,28,29,27,41,42,24)(21,34,38,30,37,36,45,46)(26,40,43,44,35,47,48,39);(17,21,33,45,22,37,23,38)(18,26,42,48,27,35,28,43)(19,30,20,34,31,46,32,36)(24,39,41,47,29,44,25,40);(1,3,4,7)(2,6,5,8)(21,37)(26,35)(30,46)(34,36)(38,45)(39,44)(40,47)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373605);