#9545605

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

4611 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452801);