#8850564

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=hi,e^2=g,f^2=g,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, 8757760);