#8893243

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

4272 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8800439);