#8914274

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

4462 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^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, 8821470);