#8917755

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

4588 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)(9,10,14,16)(11,15,19,24)(12,17,21,23)(13,18,20,22)(25,26,30,31)(27,33,36,32)(28,29,37,40)(34,35,38,39);(3,7)(5,8)(9,11,14,19)(10,15,16,24)(12,20,21,13)(17,22,23,18)(26,31)(28,37)(32,33)(34,38);(1,2)(3,5)(4,6)(7,8)(10,16)(11,19)(12,21)(18,22)(25,27,30,36)(26,32,31,33)(28,34,37,38)(29,39,40,35);(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,23)(18,24)(25,27)(26,32)(28,38)(29,35)(30,36)(31,33)(34,37)(39,40);(1,3)(2,5)(4,7)(6,8)(9,13,14,20)(10,18,16,22)(11,21,19,12)(15,23,24,17)(25,28)(26,29)(27,34)(30,37)(31,40)(32,39)(33,35)(36,38);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,24)(20,22)(21,23)(25,29)(26,28)(27,35)(30,40)(31,37)(32,38)(33,34)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,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, 8824951);