#9786756

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4532 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,e^2=g,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, 9693952);