#8987100

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

4208 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)(3,7,10,4)(5,8,13,15)(11,14,16,12)(18,23)(19,26)(25,31)(27,29)(33,34)(35,43)(36,41)(37,47)(38,39)(40,45)(42,48)(44,46);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,31)(29,30)(34,39)(36,46)(40,43)(42,47);(3,10)(4,7)(5,13)(8,15)(19,26)(20,28)(21,24)(25,31)(33,35,38,45)(34,40,39,43)(36,47,46,42)(37,41,48,44);(1,4,6,7)(2,3,9,10)(5,14,13,12)(8,16,15,11)(17,19,22,26)(18,24,23,21)(20,29,28,27)(25,30,31,32)(33,35)(34,40)(36,47)(37,44)(38,45)(39,43)(41,48)(42,46);(1,5)(2,8)(3,11)(4,12)(6,13)(7,14)(9,15)(10,16)(17,20,22,28)(18,25,23,31)(19,27,26,29)(21,30,24,32)(33,36,38,46)(34,41,39,44)(35,42,45,47)(37,43,48,40);(1,3)(2,7)(4,9)(5,11)(6,10)(8,14)(12,15)(13,16)(17,21,22,24)(18,26,23,19)(20,30,28,32)(25,29,31,27)(33,37)(34,42)(35,44)(36,43)(38,48)(39,47)(40,46)(41,45)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=ghi,e^2=g,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, 8894296);