#9889474

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	#51785	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,e^2=gh,f^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, 9796670);