#8899334

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

4449 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)(11,21)(12,18)(13,14)(15,19)(16,20)(17,24)(22,23)(25,26)(27,28)(29,37)(30,31)(32,35)(33,38)(34,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,22)(13,19)(14,23)(15,18)(16,24)(20,21)(26,31)(28,32)(33,37)(36,39);(2,5)(6,8)(9,12,16,13)(10,18,20,14)(11,22,24,19)(15,21,23,17)(25,26,30,31)(27,32,35,28)(29,33,38,37)(34,39,40,36);(3,7)(6,8)(9,13,16,12)(10,18,20,14)(11,19,24,22)(15,21,23,17)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,14,16,18)(10,13,20,12)(11,23,24,15)(17,19,21,22)(25,28,30,32)(26,27,31,35)(29,36,38,39)(33,34,37,40);(1,2)(3,6)(4,5)(7,8)(9,15,16,23)(10,19,20,22)(11,18,24,14)(12,17,13,21)(25,29,30,38)(26,33,31,37)(27,34,35,40)(28,36,32,39)

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806530);