#8917753

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

4700 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)(9,10,13,15)(11,16,19,14)(12,20,21,17)(18,22,24,23)(25,26,30,32)(27,31,35,40)(28,33,37,39)(29,34,36,38);(3,7)(5,8)(9,11,13,19)(10,14,15,16)(12,18,21,24)(17,22,20,23)(25,27,30,35)(26,31,32,40)(28,36,37,29)(33,38,39,34);(1,2)(3,5)(4,6)(7,8)(10,15)(12,21)(14,16)(18,24)(26,32)(27,35)(28,37)(34,38);(9,10)(11,14)(12,17)(13,15)(16,19)(18,22)(20,21)(23,24)(25,28)(26,33)(27,36)(29,35)(30,37)(31,38)(32,39)(34,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,22)(21,24)(25,29,30,36)(26,34,32,38)(27,37,35,28)(31,39,40,33);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,18)(13,21)(14,22)(15,20)(16,23)(19,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,38)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^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, 8824949);