#8745736

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

4369 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8652932);