#8700316

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
prime radical	#46042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4082 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(33,34,38,37)(35,40,43,39)(36,44,45,41)(42,46,48,47);(2,3)(7,9)(8,10)(15,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,34,38,37)(35,40,43,39)(36,41,45,44)(42,47,48,46);(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35)(34,39)(36,42)(37,40)(38,43)(41,46)(44,47)(45,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,46)(45,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,36,38,45)(34,41,37,44)(35,42,43,48)(39,46,40,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,37,38,34)(35,40,43,39)(36,41,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=g,e^2=gh,f^2=gi,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, 8607512);