#7648314

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

4558 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7555510);