#10521378

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

4221 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)(11,17)(12,18)(13,24)(14,19)(15,20)(16,21)(22,23)(25,26,31,34)(27,28,32,30)(29,33,37,40)(35,36,39,38);(9,11,16,14)(10,17,21,19)(12,20,22,13)(15,23,24,18)(25,27,31,32)(26,30,34,28)(29,35,37,39)(33,38,40,36);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,13)(14,20)(15,19)(16,22)(17,24)(21,23)(25,28,31,30)(26,32,34,27)(29,36,37,38)(33,39,40,35);(3,7)(6,8)(9,13,16,20)(10,15,21,24)(11,22,14,12)(17,18,19,23)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,39)(38,40);(1,3,4,7)(2,6,5,8)(9,14,16,11)(10,19,21,17)(12,20,22,13)(15,23,24,18)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,15)(10,20)(11,23)(12,19)(13,21)(14,18)(16,24)(17,22)(25,30,31,28)(26,27,34,32)(29,36,37,38)(33,39,40,35)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=i,e^2=hi,f^2=g,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, 10428574);