#9836658

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	#55706	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 9743854);