#9712830

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

4337 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)(4,6)(5,7)(9,10,16,18)(11,19,13,17)(12,22,14,15)(20,21,23,24)(25,26,32,36)(27,33,39,40)(28,29,30,34)(31,38,37,35);(1,3,5,6)(2,4,7,8)(9,11,16,13)(10,17,18,19)(12,20,14,23)(15,21,22,24)(25,27)(26,33)(28,38)(29,37)(30,35)(31,34)(32,39)(36,40);(9,12,16,14)(10,15,18,22)(11,20,13,23)(17,21,19,24)(25,28,32,30)(26,34,36,29)(27,35,39,38)(31,33,37,40);(9,13,16,11)(10,17,18,19)(12,20,14,23)(15,24,22,21)(25,29,32,34)(26,28,36,30)(27,37,39,31)(33,38,40,35);(1,2)(3,4)(5,7)(6,8)(9,14,16,12)(10,15,18,22)(11,20,13,23)(17,24,19,21)(25,30,32,28)(26,34,36,29)(27,38,39,35)(31,33,37,40);(1,4)(2,6)(3,7)(5,8)(9,15)(10,14)(11,21)(12,18)(13,24)(16,22)(17,23)(19,20)(25,31,32,37)(26,35,36,38)(27,34,39,29)(28,33,30,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=h,e^2=h,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, 9620026);