#10020557

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

4508 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)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,18,17)(12,20,22,16)(19,24,21,23)(25,26,30,33)(27,31,35,29)(28,32,36,39)(34,37,40,38);(2,5)(6,7)(9,10,13,14)(11,17,18,15)(12,16,22,20)(19,24,21,23)(25,27,30,35)(26,29,33,31)(28,34,36,40)(32,38,39,37);(2,5)(6,7)(10,14)(11,18)(16,20)(19,21)(28,36)(32,39)(34,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,37)(36,40)(38,39);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,22)(20,23)(25,28)(26,32)(27,34)(29,38)(30,36)(31,37)(33,39)(35,40);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,29,30,31)(26,27,33,35)(28,37,36,38)(32,40,39,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9927753);