#8115502

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

4287 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8022698);