#8750646

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

4105 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8657842);