#8097778

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

4418 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:

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

Polycyclic presentation:

2^9|c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004974);