#8231441

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

4315 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=g,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, 8138637);