#8527526

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

4429 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)(9,10,14,18)(11,15,21,16)(12,13,17,23)(19,24,22,20)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(3,7)(6,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,18)(17,19)(23,24)(26,28)(27,33)(32,36)(35,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,16,18,15)(12,19,17,22)(13,20,23,24)(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)(10,13)(11,19)(14,17)(15,24)(16,20)(18,23)(21,22)(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)(10,16)(12,19)(13,20)(14,21)(15,18)(17,22)(23,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(9,13)(10,17)(11,20)(12,18)(14,23)(15,19)(16,22)(21,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8434722);