#8258055

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

4302 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=gi,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, 8165251);