#8596415

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

4034 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8503611);