#10464594

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

4994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371790);