#8841699

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

4297 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8748895);