#8158656

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	#53483	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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:

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

Polycyclic presentation:

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

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, 8065852);