#9054444

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961640);