#9720438

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9627634);