#8824119

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) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=g,d^2=hi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731315);