#8636229

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

4542 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=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, 8543425);