#6468240

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1614 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)(17,18,24,40,23,29,22,28)(19,30,39,25,34,48,33,43)(20,26,38,47,36,45,21,27)(31,46,32,41,35,42,37,44);(10,14)(11,12)(17,19,23,34)(18,25,29,43)(20,35,36,31)(21,37,38,32)(22,39,24,33)(26,44,45,41)(27,46,47,42)(28,48,40,30);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,23,36)(18,26,29,45)(19,31,34,35)(21,24,38,22)(25,41,43,44)(27,40,47,28)(30,46,48,42)(32,33,37,39);(9,12,13,11)(10,16,14,15)(17,21,23,38)(18,27,29,47)(19,32,34,37)(20,22,36,24)(25,42,43,46)(26,28,45,40)(30,41,48,44)(31,39,35,33);(1,3,4,7)(2,6,5,8)(20,36)(21,38)(26,45)(27,47)(31,35)(32,37)(41,44)(42,46)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,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, 6375436);