#8739205

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

4200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8646401);