#8197938

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	#52164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4393 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8105134);