#8196386

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

3913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103582);