#9027212

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

4056 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=g,e^2=gh,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, 8934408);