#6485313

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1559 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)(17,18,22,27,23,28,20,25)(19,29,30,39,32,42,37,24)(21,34,35,40,36,26,33,41)(31,45,44,38,47,43,46,48);(9,10)(11,16)(12,15)(13,14)(17,19,23,32)(18,24,28,39)(20,30,22,37)(21,31,36,47)(25,29,27,42)(26,38,34,48)(33,44,35,46)(40,43,41,45);(2,5)(6,8)(10,14)(15,16)(17,20,23,22)(18,25,28,27)(19,30,32,37)(21,35,36,33)(24,29,39,42)(26,41,34,40)(31,46,47,44)(38,45,48,43);(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,31)(20,33)(22,35)(23,36)(24,38)(25,40)(27,41)(28,34)(29,43)(30,44)(32,47)(37,46)(39,48)(42,45);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(21,36)(26,34)(31,47)(33,35)(38,48)(40,41)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6392509);