#7647024

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=h,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, 7554220);