#10055583

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

4380 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)(3,7)(4,6)(5,8)(9,10,14,17)(11,12,15,18)(13,16,22,24)(19,20,23,21)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,2,4,6)(3,5,8,7)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(27,33)(30,37)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(11,15)(12,18)(19,23)(20,21)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,21)(20,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(9,12,14,18)(10,15,17,11)(13,21,22,20)(16,19,24,23)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9962779);