#8745912

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

4499 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=hi,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, 8653108);