#9023530

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

4486 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,d^2=g,f^2=gi,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, 8930726);