#8754853

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	#53697	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662049);