#9545780

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

4235 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452976);