#8912168

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

4518 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)(11,16,20,15)(12,21,22,13)(18,19,23,24)(25,26,30,35)(27,31,37,39)(28,33,38,32)(29,36,40,34);(2,5)(6,7)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,24)(22,23)(25,27,30,37)(26,31,35,39)(28,34,38,36)(29,32,40,33);(1,2)(3,6)(4,5)(7,8)(12,22)(13,21)(18,23)(19,24)(25,28,30,38)(26,32,35,33)(27,36,37,34)(29,39,40,31);(1,3,4,8)(2,6,5,7)(9,12)(10,13)(11,18)(14,22)(15,19)(16,24)(17,21)(20,23)(25,28)(26,33)(27,34)(29,31)(30,38)(32,35)(36,37)(39,40);(9,11)(10,16)(12,23)(13,19)(14,20)(15,17)(18,22)(21,24)(25,27)(26,31)(28,36)(29,32)(30,37)(33,40)(34,38)(35,39);(9,13)(10,12)(11,19)(14,21)(15,18)(16,23)(17,22)(20,24)(25,29,30,40)(26,34,35,36)(27,32,37,33)(28,31,38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8819364);