#8662909

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

4601 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570105);