#8968087

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

4336 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=gi,e^2=h,f^2=hi,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, 8875283);