#9843552

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

4392 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,e^2=gi,f^2=gh,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, 9750748);