#9683990

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

4304 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,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, 9591186);