#6484967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1392 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,28)(19,29,34,45)(20,26,36,25)(21,27,22,38)(24,32,40,33)(30,46,31,44)(35,42,37,41)(39,47,43,48);(2,5)(6,7)(10,14)(11,12)(17,19,21,32,23,34,22,33)(18,24,38,45,28,40,27,29)(20,30,37,48,36,31,35,47)(25,39,41,46,26,43,42,44);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,23,36)(18,25,28,26)(19,30,34,31)(21,37,22,35)(24,39,40,43)(27,42,38,41)(29,44,45,46)(32,48,33,47);(9,12,13,11)(10,15,14,16)(17,20)(18,26)(19,31)(21,37)(22,35)(23,36)(24,39)(25,28)(27,41)(29,44)(30,34)(32,47)(33,48)(38,42)(40,43)(45,46);(10,14)(15,16)(17,21,23,22)(18,27,28,38)(19,32,34,33)(20,35,36,37)(24,29,40,45)(25,41,26,42)(30,47,31,48)(39,46,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6392163);