#8839565

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

4152 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=ghi,e^2=g,f^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, 8746761);