#9714552

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

4353 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,28)(29,37)(30,31)(32,34)(33,39)(35,38)(36,40);(1,3)(2,6)(4,7)(5,8)(11,17)(15,21)(18,23)(22,24)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,38)(39,40);(10,14)(15,21)(16,19)(22,24)(26,31)(27,34)(33,37)(35,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,38)(36,37)(39,40);(1,4,5,6)(2,3,7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,40,37,35);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,39)(26,33,31,37)(27,35,34,40)(28,36,32,38)

Polycyclic presentation:

2^9|e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9621748);