#8115516

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3711 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=h,e^2=h,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, 8022712);