#8228631

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

4569 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=hi,e^2=hi,f^2=g,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, 8135827);