#8402373

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

4445 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309569);