#8689964

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

4632 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8597160);