#8987127

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

4593 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,15,18,12)(13,16,21,23)(19,22,24,20)(25,26,31,36)(27,37,38,32)(28,29,34,33)(30,35,39,40);(3,8)(6,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27,31,38)(26,32,36,37)(28,35,34,40)(29,30,33,39);(1,3,5,8)(2,6,4,7)(11,18)(12,15)(13,21)(16,23)(28,34)(29,33)(30,39)(35,40);(3,8)(6,7)(9,12,14,15)(10,11,17,18)(13,22,21,20)(16,24,23,19)(25,28)(26,33)(27,35)(29,36)(30,37)(31,34)(32,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,29,31,33)(26,34,36,28)(27,30,38,39)(32,40,37,35);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,30)(26,35)(27,33)(28,37)(29,38)(31,39)(32,34)(36,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=hi,e^2=g,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, 8894323);