#8662930

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

4103 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570126);