#9923664

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

4602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830860);