#8864450

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

4106 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=i,e^2=gi,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, 8771646);