#8416909

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324105);