#9923655

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

4566 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,14)(13,16)(15,17)(18,24)(20,23)(21,22)(25,26)(27,33)(28,37)(29,31)(30,34)(32,35)(36,38)(39,40);(2,5)(6,7)(9,11,15,13)(10,16,17,19)(12,20,18,22)(14,21,24,23)(25,27,32,29)(26,33,35,31)(28,36,30,39)(34,40,37,38);(1,2)(3,6)(4,5)(7,8)(10,17)(12,18)(16,19)(20,22)(25,28,32,30)(26,34,35,37)(27,36,29,39)(31,38,33,40);(1,3,4,8)(2,6,5,7)(9,12,15,18)(10,14,17,24)(11,20,13,22)(16,21,19,23)(25,29,32,27)(26,31,35,33)(28,36,30,39)(34,40,37,38);(9,13,15,11)(10,16,17,19)(12,22,18,20)(14,21,24,23)(25,30,32,28)(26,34,35,37)(27,36,29,39)(31,40,33,38);(9,14)(10,18)(11,21)(12,17)(13,23)(15,24)(16,22)(19,20)(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=ghi,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, 9830851);