#9937805

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

4236 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9845001);