#8581800

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

4116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488996);