#9844037

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

4482 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)(11,19)(12,15)(13,17)(14,23)(16,21)(18,24)(20,22)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,40)(36,37);(1,2,5,7)(3,4,8,6)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,18,22,23)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(10,12)(16,19)(17,22)(18,23)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,40,38,36);(9,12,15,10)(11,19,21,16)(13,22,20,17)(14,23,24,18)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(1,4,5,6)(2,3,7,8)(9,13)(10,17)(11,14)(12,22)(15,20)(16,18)(19,23)(21,24)(25,29,31,27)(26,32,34,28)(30,35,39,37)(33,36,38,40);(1,3)(2,4)(5,8)(6,7)(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,22)(17,19)(25,30,31,39)(26,33,34,38)(27,35,29,37)(28,36,32,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=h,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, 9751233);