#9070708

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
prime radical	#35684	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,e^2=g,f^2=g,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, 8977904);