#8772981

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

4214 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=h,e^2=gh,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, 8680177);