#9064150

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
prime radical	#43107	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 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:

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

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971346);