#9061415

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	#43367	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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)(11,16)(12,18)(13,19)(14,22)(15,17)(20,24)(21,23)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,37,40,33);(1,2,5,6)(3,7,8,4)(9,11)(10,16)(12,15)(13,20)(14,21)(17,18)(19,24)(22,23)(25,27,30,36)(26,31,34,35)(28,33,38,37)(29,32,40,39);(1,2)(3,7)(4,8)(5,6)(10,17)(11,12)(13,23)(14,24)(25,27,30,36)(26,31,34,35)(28,37,38,33)(29,39,40,32);(2,6)(3,8)(9,12,15,11)(10,18,17,16)(13,20,23,22)(14,19,24,21)(25,27)(26,31)(28,37)(29,32)(30,36)(33,38)(34,35)(39,40);(1,3)(2,4)(5,8)(6,7)(9,13,15,23)(10,19,17,21)(11,20,12,22)(14,16,24,18)(25,28,30,38)(26,32,34,39)(27,33,36,37)(29,31,40,35);(1,4)(2,3)(5,7)(6,8)(9,14)(10,20)(11,21)(12,19)(13,16)(15,24)(17,22)(18,23)(25,29,30,40)(26,33,34,37)(27,32,36,39)(28,31,38,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968611);