#6485440

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,23,27)(19,28,32,42)(20,25,22,36)(21,26,34,40)(24,29,38,31)(30,41,44,48)(33,39,35,46)(37,43,47,45);(9,10)(11,16)(12,15)(13,14)(17,19,20,29,23,32,22,31)(18,24,36,42,27,38,25,28)(21,30,33,43,34,44,35,45)(26,37,46,48,40,47,39,41);(2,5)(6,8)(10,14)(15,16)(17,20,23,22)(18,25,27,36)(19,29,32,31)(21,33,34,35)(24,28,38,42)(26,39,40,46)(30,43,44,45)(37,41,47,48);(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,30)(20,33)(22,35)(23,34)(24,37)(25,39)(27,40)(28,41)(29,43)(31,45)(32,44)(36,46)(38,47)(42,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(21,34)(26,40)(30,44)(33,35)(37,47)(39,46)(41,48)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392636);