#8106012

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

4598 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013208);