#8810244

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	#36493	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4463 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717440);