#9844024

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

4544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,20)(12,14)(13,16)(15,17)(18,19)(21,24)(22,23)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,40)(36,37);(1,2,5,7)(3,4,8,6)(9,11,15,13)(10,16,17,20)(12,21,19,23)(14,22,18,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(10,17)(12,19)(16,20)(21,23)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,40,38,36);(9,12,15,19)(10,18,17,14)(11,21,13,23)(16,24,20,22)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(1,4,5,6)(2,3,7,8)(9,13,15,11)(10,16,17,20)(12,23,19,21)(14,22,18,24)(25,29,31,27)(26,32,34,28)(30,35,39,37)(33,36,38,40);(1,3)(2,4)(5,8)(6,7)(9,14,15,18)(10,19,17,12)(11,22,13,24)(16,23,20,21)(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=gi,f^2=h,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, 9751220);