#8695996

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

4226 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,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, 8603192);