#8772863

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

4147 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,e^2=h,f^2=gh,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, 8680059);