#8621982

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=g,e^2=i,f^2=h,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, 8529178);