#8821686

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

4052 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8728882);