#9844110

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

4461 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=gh,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, 9751306);