#9836062

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

4499 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=i,f^2=g,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, 9743258);