#8924381

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

4482 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)(3,5,7,8)(9,10)(11,17)(12,22)(13,15)(14,18)(16,19)(20,24)(21,23)(25,26)(27,34)(28,33)(29,32)(30,35)(31,36)(37,38)(39,40);(1,3,4,7)(2,5,6,8)(9,11,16,13)(10,17,19,15)(12,21,14,20)(18,24,22,23)(25,27,31,29)(26,32,36,34)(28,30,37,39)(33,40,38,35);(1,2)(3,5)(4,6)(7,8)(9,12,16,14)(10,18,19,22)(11,20,13,21)(15,24,17,23)(25,28,31,37)(26,33,36,38)(27,30,29,39)(32,40,34,35);(9,13,16,11)(10,15,19,17)(12,20,14,21)(18,23,22,24)(25,29,31,27)(26,34,36,32)(28,30,37,39)(33,40,38,35);(1,3)(2,5)(4,7)(6,8)(9,14,16,12)(10,18,19,22)(11,21,13,20)(15,24,17,23)(25,30)(26,35)(27,37)(28,29)(31,39)(32,33)(34,38)(36,40);(1,2)(3,5)(4,6)(7,8)(9,15)(10,11)(12,23)(13,19)(14,24)(16,17)(18,21)(20,22)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,37)(39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,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, 8831577);