#8271191

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

4579 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,32)(28,34)(29,36)(30,35)(31,33)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,27)(26,32)(28,30)(29,33)(31,36)(34,35)(37,38)(39,40);(1,2,5,4)(3,6,8,7)(12,20)(15,22)(18,24)(21,23)(26,33)(27,36)(30,37)(34,39);(1,2,5,4)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,28)(26,34)(27,37)(29,35)(30,36)(31,38)(32,40)(33,39);(1,4,5,2)(3,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,29)(26,27)(28,35)(30,39)(31,32)(33,36)(34,37)(38,40);(1,3)(2,7)(4,6)(5,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,30,31,37)(26,35,33,40)(27,28,36,38)(29,39,32,34)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8178387);