#10482497

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

4518 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)(9,10,15,12)(11,19,21,16)(13,17,20,22)(14,18,24,23)(25,26,31,35)(27,33,34,30)(28,37,29,32)(36,38,39,40);(1,3,5,8)(2,6,7,4)(9,11,15,21)(10,16,12,19)(13,14,20,24)(17,23,22,18)(25,27)(26,30)(28,36)(29,39)(31,34)(32,38)(33,35)(37,40);(1,3)(2,4)(5,8)(6,7)(25,28,31,29)(26,32,35,37)(27,36,34,39)(30,38,33,40);(1,4)(2,3)(5,6)(7,8)(9,12,15,10)(11,16,21,19)(13,22,20,17)(14,23,24,18)(25,29,31,28)(26,32,35,37)(27,36,34,39)(30,40,33,38);(9,13)(10,17)(11,14)(12,22)(15,20)(16,23)(18,19)(21,24)(25,27)(26,33)(28,36)(29,39)(30,35)(31,34)(32,40)(37,38);(9,14)(10,18)(11,20)(12,23)(13,21)(15,24)(16,17)(19,22)(25,30,31,33)(26,34,35,27)(28,38,29,40)(32,39,37,36)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gi,f^2=h,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, 10389693);