#8973120

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

4073 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=g,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, 8880316);