#9875498

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	#44417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9782694);