#9015552

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

4413 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,e^2=gi,f^2=h,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, 8922748);