#9017760

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

4143 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=hi,e^2=ghi,f^2=gh,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, 8924956);