#8115338

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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, 8022534);