#7656040

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

4116 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)(11,13,17,15)(12,20,22,16)(19,23,24,21)(25,26,31,32)(27,36,33,28)(29,34,30,35)(37,40,38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(26,32)(27,33)(34,35)(37,38);(9,12,14,22)(10,16,18,20)(11,19,17,24)(13,21,15,23)(25,27,31,33)(26,28,32,36)(29,37,30,38)(34,39,35,40);(25,28)(26,33)(27,32)(29,39)(30,40)(31,36)(34,38)(35,37);(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,23)(22,24)(25,29,31,30)(26,34,32,35)(27,37,33,38)(28,39,36,40);(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,20)(22,23)(25,30,31,29)(26,35,32,34)(27,38,33,37)(28,40,36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7563236);