#9923651

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

4316 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,21)(13,18)(14,22)(15,17)(19,23)(20,24)(25,26)(27,33)(28,37)(29,31)(30,34)(32,35)(36,38)(39,40);(2,5)(6,7)(9,11,15,19)(10,16,17,23)(12,20,22,13)(14,18,21,24)(25,27,32,29)(26,33,35,31)(28,36,30,39)(34,40,37,38);(10,17)(11,19)(12,22)(18,24)(25,28,32,30)(26,34,35,37)(27,36,29,39)(31,38,33,40);(9,12)(10,14)(11,20)(13,19)(15,22)(16,18)(17,21)(23,24)(25,29,32,27)(26,31,35,33)(28,36,30,39)(34,40,37,38);(1,2)(3,7)(4,5)(6,8)(9,13,15,20)(10,18,17,24)(11,12,19,22)(14,16,21,23)(25,30,32,28)(26,34,35,37)(27,36,29,39)(31,40,33,38);(1,3)(2,6)(4,8)(5,7)(9,14)(10,12)(11,18)(13,23)(15,21)(16,20)(17,22)(19,24)(25,31)(26,27)(28,38)(29,35)(30,40)(32,33)(34,36)(37,39)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830847);