#8115307

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

4533 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022503);