#6485740

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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

1390 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)(11,15)(12,16)(13,14)(17,18,23,29)(19,30,34,40)(20,26,37,42)(21,27,22,28)(24,32,41,33)(25,36,43,35)(31,45,48,38)(39,47,44,46);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,23,34)(18,24,29,41)(20,31,37,48)(21,32,22,33)(25,38,43,45)(26,39,42,44)(27,30,28,40)(35,46,36,47);(10,14)(15,16)(17,20,23,37)(18,25,29,43)(19,31,34,48)(21,35,22,36)(24,38,41,45)(26,27,42,28)(30,44,40,39)(32,46,33,47);(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,36)(22,35)(23,37)(24,39)(25,28)(27,43)(29,42)(30,45)(32,47)(33,46)(34,48)(38,40)(41,44);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,27,29,28)(19,32,34,33)(20,35,37,36)(24,30,41,40)(25,42,43,26)(31,46,48,47)(38,44,45,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi,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, 6392936);