#9028704

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

4468 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)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,40)(38,39);(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(27,36)(29,37)(32,35)(34,40);(1,3,5,8)(2,6,4,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,31,36)(26,32,30,35)(28,29,39,37)(33,40,38,34);(3,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,28,31,39)(26,33,30,38)(27,37,36,29)(32,34,35,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,24)(20,22)(21,23)(25,29,31,37)(26,34,30,40)(27,28,36,39)(32,38,35,33);(1,4,5,2)(3,6,8,7)(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,30,31,26)(27,32,36,35)(28,38,39,33)(29,40,37,34)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935900);