#9729240

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	#54364	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4168 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,e^2=h,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, 9636436);