#7645856

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

4370 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)(10,11)(13,21)(15,17)(19,24)(26,32)(30,34)(33,36)(35,40);(1,2)(3,6)(4,5)(7,8)(25,26,31,32)(27,36,38,33)(28,34,37,30)(29,35,39,40);(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27,31,38)(26,33,32,36)(28,29,37,39)(30,35,34,40);(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28)(26,34)(27,29)(30,32)(31,37)(33,40)(35,36)(38,39);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29)(26,35)(27,37)(28,38)(30,36)(31,39)(32,40)(33,34);(1,3,4,7)(2,6,5,8)(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,34)(26,28,32,37)(27,35,38,40)(29,36,39,33)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=h,e^2=h,f^2=g,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, 7553052);