#8718237

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

4593 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)(3,8)(4,5)(6,7)(9,10,15,19)(11,16,20,23)(12,17,21,22)(13,14,18,24)(25,26)(27,32)(28,37)(29,34)(30,31)(33,38)(35,36)(39,40);(1,3,6,5)(2,4,7,8)(9,11,15,20)(10,16,19,23)(12,13,21,18)(14,22,24,17)(25,26)(27,35)(28,33)(29,34)(30,31)(32,36)(37,38)(39,40);(1,3,6,5)(2,4,7,8)(11,20)(12,21)(16,23)(17,22)(26,31)(27,36)(33,37)(34,40);(2,7)(3,5)(9,12,15,21)(10,17,19,22)(11,13,20,18)(14,23,24,16)(25,27)(26,32)(28,34)(29,37)(30,36)(31,35)(33,39)(38,40);(1,4,6,8)(2,3,7,5)(9,13)(10,14)(11,21)(12,20)(15,18)(16,22)(17,23)(19,24)(25,28)(26,33)(27,34)(29,35)(30,38)(31,37)(32,39)(36,40);(1,5,6,3)(2,4,7,8)(9,14)(10,18)(11,22)(12,16)(13,19)(15,24)(17,20)(21,23)(25,29,30,39)(26,34,31,40)(27,33,36,37)(28,32,38,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8625433);