#8992087

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
prime radical	#52417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4304 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,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, 8899283);