#9061469

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4500 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)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,30,31)(27,28,35,32)(29,37,39,33)(34,40,38,36);(1,2,5,6)(3,7,8,4)(9,10,14,18)(11,19,20,15)(12,13,17,16)(21,22,23,24)(26,31)(28,32)(33,37)(36,40);(1,2)(3,7)(4,8)(5,6)(11,20)(12,17)(13,16)(15,19)(25,27,30,35)(26,32,31,28)(29,38,39,34)(33,36,37,40);(2,6)(3,8)(9,11)(10,15)(12,23)(13,22)(14,20)(16,24)(17,21)(18,19)(25,28,30,32)(26,27,31,35)(29,36,39,40)(33,34,37,38);(1,3)(2,4)(5,8)(6,7)(9,12,14,17)(10,16,18,13)(11,21,20,23)(15,22,19,24)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,16)(10,17,18,12)(11,22,20,24)(15,21,19,23)(25,29,30,39)(26,33,31,37)(27,34,35,38)(28,36,32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,e^2=gh,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, 8968665);