#8839083

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

4316 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)(3,7,9,4)(5,13,15,10)(11,12,14,16)(18,20)(19,25)(24,28)(26,31)(33,34,38,40)(35,36,43,39)(37,45,47,41)(42,48,46,44);(1,3)(2,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,35,38,43)(34,39,40,36)(37,46,47,42)(41,44,45,48);(2,8)(3,9)(10,13)(11,14)(19,25)(23,27)(26,31)(30,32)(34,40)(36,39)(41,45)(44,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,16)(12,13)(14,15)(17,19,22,25)(18,23,20,27)(21,26,29,31)(24,30,28,32)(33,36)(34,35)(37,44)(38,39)(40,43)(41,42)(45,46)(47,48);(1,4,6,7)(2,9,8,3)(5,12,15,16)(10,14,13,11)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,35)(34,39)(36,40)(37,42)(38,43)(41,48)(44,45)(46,47);(1,5,6,15)(2,10,8,13)(3,11,9,14)(4,12,7,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,37)(34,41)(35,42)(36,44)(38,47)(39,48)(40,45)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8746279);