#8492939

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

4079 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)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,23,21)(19,26,24,20)(22,25,31,29)(27,30,32,28)(33,34,39,43)(35,44,48,40)(36,37,47,41)(38,42,45,46);(2,3)(7,9)(8,10)(14,15)(18,21)(20,26)(25,29)(28,30)(33,35)(34,40)(36,45)(37,42)(38,47)(39,48)(41,46)(43,44);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32)(33,34,39,43)(35,40,48,44)(36,41,47,37)(38,42,45,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,26)(18,24,21,19)(22,30,31,28)(25,27,29,32)(33,36)(34,41)(35,45)(37,43)(38,48)(39,47)(40,46)(42,44);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,18)(19,26,24,20)(22,29,31,25)(27,30,32,28)(33,37,39,41)(34,36,43,47)(35,46,48,42)(38,44,45,40);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,31)(18,25,21,29)(19,27,24,32)(20,28,26,30)(33,38,39,45)(34,42,43,46)(35,47,48,36)(37,44,41,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8400135);