#9065982

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

4299 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,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, 8973178);