#9065296

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

4376 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)(11,15)(12,17)(13,20)(14,16)(18,24)(19,22)(21,23)(25,26)(27,33)(28,29)(30,38)(31,32)(34,39)(35,37)(36,40);(1,2)(3,4)(5,6)(7,8)(9,11,14,19)(10,15,16,22)(12,20,23,18)(13,21,24,17)(26,32)(29,33)(34,38)(37,40);(2,6)(3,7)(10,16)(11,19)(17,21)(18,20)(25,27,31,28)(26,29,32,33)(30,35,39,36)(34,37,38,40);(3,7)(4,8)(9,12)(10,17)(11,20)(13,15)(14,23)(16,21)(18,19)(22,24)(25,28,31,27)(26,33,32,29)(30,35,39,36)(34,37,38,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,21)(14,19)(16,22)(17,24)(20,23)(25,29,31,33)(26,28,32,27)(30,37,39,40)(34,36,38,35);(1,4)(2,3)(5,8)(6,7)(9,13,14,24)(10,18,16,20)(11,21,19,17)(12,22,23,15)(25,30,31,39)(26,34,32,38)(27,35,28,36)(29,37,33,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,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, 8972492);