#8137588

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

4344 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044784);