#8718462

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

4649 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8625658);