#8093142

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

4554 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)(6,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,31,32)(27,29,34,39)(28,35,30,33)(36,40,37,38);(2,5)(6,8)(12,20)(15,22)(18,24)(21,23)(26,32)(29,39)(33,35)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,31,34)(26,29,32,39)(28,36,30,37)(33,38,35,40);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,28,31,30)(26,33,32,35)(27,36,34,37)(29,38,39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,29)(26,34)(27,32)(28,38)(30,40)(31,39)(33,37)(35,36);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,30,31,28)(26,35,32,33)(27,37,34,36)(29,40,39,38)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000338);