#9889527

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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)(11,14)(12,13)(15,23)(16,18)(17,24)(19,20)(21,22)(25,26,31,36)(27,30,35,32)(28,33,29,34)(37,38,39,40);(1,2,5,7)(3,8,4,6)(9,11,16,13)(10,14,18,12)(15,19,24,21)(17,22,23,20)(25,27,31,35)(26,32,36,30)(28,37,29,39)(33,40,34,38);(1,3,5,4)(2,6,7,8)(9,12,16,14)(10,11,18,13)(15,20,24,22)(17,19,23,21)(25,28,31,29)(26,33,36,34)(27,37,35,39)(30,38,32,40);(9,13,16,11)(10,12,18,14)(15,21,24,19)(17,20,23,22)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,40)(38,39);(1,2)(3,6)(4,8)(5,7)(9,14,16,12)(10,11,18,13)(15,20,24,22)(17,21,23,19)(25,29,31,28)(26,34,36,33)(27,37,35,39)(30,38,32,40);(1,4,5,3)(2,6,7,8)(9,15)(10,17)(11,19)(12,20)(13,21)(14,22)(16,24)(18,23)(25,30,31,32)(26,35,36,27)(28,38,29,40)(33,39,34,37)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=h,e^2=gh,f^2=ghi,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, 9796723);