#8962984

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

4229 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)(3,8)(4,9)(5,10)(6,7)(11,14)(12,16)(13,15)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,34,39,44)(35,42,46,40)(36,41,37,47)(38,43,48,45);(1,3,7,11)(2,8,6,14)(4,12,13,5)(9,16,15,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,40)(36,43)(37,45)(38,41)(39,46)(42,44)(47,48);(2,6)(3,11)(5,12)(9,15)(19,25)(21,29)(23,26)(24,28)(33,36,39,37)(34,41,44,47)(35,43,46,45)(38,42,48,40);(1,4)(2,9)(3,12)(5,11)(6,15)(7,13)(8,16)(10,14)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37,39,36)(34,41,44,47)(35,45,46,43)(38,42,48,40);(1,5,7,12)(2,10,6,16)(3,13,11,4)(8,15,14,9)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,35)(34,42)(36,45)(37,43)(38,41)(39,46)(40,44)(47,48);(1,6,7,2)(3,14,11,8)(4,15,13,9)(5,16,12,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,38)(34,43)(35,41)(36,40)(37,42)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8870180);