#9027021

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

4052 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)(3,5,9,7)(4,8,12,16)(11,14,13,15)(17,18)(19,24)(20,25)(21,30)(22,23)(26,32)(27,31)(28,29)(33,34,38,41)(35,39,43,37)(36,40,46,48)(42,47,44,45);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,28)(21,26)(22,23)(25,29)(27,31)(30,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(3,9)(4,12)(5,7)(8,16)(18,23)(19,27)(20,29)(21,32)(33,35,38,43)(34,39,41,37)(36,44,46,42)(40,45,48,47);(1,3,6,9)(2,7,10,5)(4,13,12,11)(8,14,16,15)(17,19,22,27)(18,24,23,31)(20,30,29,26)(21,28,32,25)(33,35)(34,39)(36,44)(37,41)(38,43)(40,45)(42,46)(47,48);(1,4,6,12)(2,8,10,16)(3,11,9,13)(5,14,7,15)(17,20)(18,25)(19,26)(21,31)(22,29)(23,28)(24,32)(27,30)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,5,6,7)(2,9,10,3)(4,14,12,15)(8,13,16,11)(17,21,22,32)(18,26,23,30)(19,28,27,25)(20,31,29,24)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,48,44)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8934217);