#9028407

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

4405 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8935603);