#8696029

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

4018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603225);