#8093584

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

4697 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000780);