#9947440

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	#59023	C₂
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854636);