#9938177

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

4321 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845373);