#9715477

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

Polycyclic presentation:

2^9|a^2=gh,e^2=gi,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, 9622673);