#9848668

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

4343 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^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=eghi,d^c=dh,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, 9755864);