#8898945

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

4516 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)(3,4)(5,6)(7,8)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26,31,30)(27,33,28,32)(29,34,38,37)(35,40,36,39);(2,6)(4,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,27,31,28)(26,32,30,33)(29,35,38,36)(34,39,37,40);(1,2,5,6)(3,7,8,4)(10,15)(14,17)(16,23)(22,24)(25,27,31,28)(26,33,30,32)(29,35,38,36)(34,40,37,39);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,28,31,27)(26,33,30,32)(29,35,38,36)(34,39,37,40);(1,4,5,7)(2,3,6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,31,38)(26,34,30,37)(27,35,28,36)(32,39,33,40);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,30,31,26)(27,33,28,32)(29,37,38,34)(35,40,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806141);