#8284433

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

4253 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8191629);