#8605689

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=g,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, 8512885);