#9720512

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

4909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,e^2=ghi,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, 9627708);