#9843860

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

4409 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=gh,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9751056);