#8750321

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

4505 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657517);