#9064897

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

4610 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8972093);