#9716272

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
prime radical	#59097	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4453 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)(3,6)(4,7)(5,8)(9,10)(11,20)(12,13)(14,24)(15,17)(16,21)(18,23)(19,22)(25,26)(27,35)(28,33)(29,39)(30,32)(31,36)(34,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,21)(10,16,17,20)(12,22,23,14)(13,24,18,19)(25,27,30,36)(26,31,32,35)(28,37,38,29)(33,39,40,34);(10,17)(11,21)(12,23)(19,24)(26,32)(31,35)(33,40)(34,39);(9,12)(10,18)(11,22)(13,17)(14,21)(15,23)(16,19)(20,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,18)(10,12,17,23)(11,19,21,24)(14,20,22,16)(25,28,30,38)(26,33,32,40)(27,29,36,37)(31,34,35,39);(1,2)(3,5)(4,7)(6,8)(9,14,15,22)(10,19,17,24)(11,12,21,23)(13,20,18,16)(25,29)(26,34)(27,28)(30,37)(31,33)(32,39)(35,40)(36,38)

Polycyclic presentation:

2^9|b^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, 9623468);