#8599843

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

4379 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8507039);