#9815001

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

4475 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)(4,7)(5,8)(9,10,14,16)(11,15,19,23)(12,17,22,21)(13,20,24,18)(25,26)(27,35)(28,31)(29,33)(30,39)(32,37)(34,40)(36,38);(1,2,5,8)(3,4,7,6)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,24)(18,22)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,34,38,39);(10,16)(11,19)(17,21)(18,20)(26,28)(32,35)(33,38)(34,39);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,24)(18,22)(25,28,31,26)(27,35,37,32)(29,38,36,33)(30,39,40,34);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,20)(13,23)(14,22)(15,24)(16,21)(18,19)(25,29)(26,33)(27,30)(28,38)(31,36)(32,34)(35,39)(37,40);(1,4,5,6)(2,7,8,3)(9,13,14,24)(10,18,16,20)(11,21,19,17)(12,15,22,23)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=hi,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, 9722197);