#8943760

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

4311 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8850956);