#8384950

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

4118 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)(3,11,13,8)(4,12,14,9)(5,10,16,15)(18,23)(19,26)(25,30)(27,32);(2,7)(4,14)(8,11)(10,15)(17,18,22,23)(19,20,26,24)(21,25,29,30)(27,28,32,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(2,7)(5,16)(8,11)(9,12)(21,29)(25,30)(27,32)(28,31)(34,38)(39,41)(40,44)(47,48);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,12,16,9)(17,19,22,26)(18,24,23,20)(21,27,29,32)(25,31,30,28)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4)(2,9)(3,10)(5,11)(6,14)(7,12)(8,16)(13,15)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,5,6,16)(2,10,7,15)(3,12,13,9)(4,11,14,8)(17,21,22,29)(18,25,23,30)(19,27,26,32)(20,28,24,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8292146);