#9836609

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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)(11,14)(12,13)(15,23)(16,18)(17,24)(19,20)(21,22)(26,32)(28,36)(33,38)(35,37);(1,2)(3,7)(4,5)(6,8)(9,11,16,13)(10,14,18,12)(15,19,24,21)(17,22,23,20)(25,26,31,32)(27,33,29,38)(28,34,36,39)(30,35,40,37);(1,3)(2,6)(4,7)(5,8)(9,12,16,14)(10,11,18,13)(15,20,24,22)(17,19,23,21)(25,27,31,29)(26,33,32,38)(28,35,36,37)(30,34,40,39);(9,13,16,11)(10,12,18,14)(15,21,24,19)(17,20,23,22)(25,28,31,36)(26,34,32,39)(27,35,29,37)(30,33,40,38);(1,4,5,2)(3,7,8,6)(9,14,16,12)(10,11,18,13)(15,20,24,22)(17,21,23,19)(25,29,31,27)(26,33,32,38)(28,37,36,35)(30,34,40,39);(1,3)(2,7)(4,6)(5,8)(9,15)(10,17)(11,19)(12,20)(13,21)(14,22)(16,24)(18,23)(25,30,31,40)(26,35,32,37)(27,34,29,39)(28,33,36,38)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gh,e^2=gi,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, 9743805);