#9718563

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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)(9,10,14,15)(11,17,19,12)(13,16,22,23)(18,24,21,20)(26,30)(31,37)(32,38)(39,40);(2,6)(3,7)(9,11)(10,12)(13,21)(14,19)(15,17)(16,24)(18,22)(20,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,3)(2,4)(5,7)(6,8)(10,15)(12,17)(16,23)(20,24);(1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(9,11)(10,12)(13,21)(14,19)(15,17)(16,24)(18,22)(20,23)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9625759);