#7646990

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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:

(1,2,5,4)(3,7,8,6)(11,12)(14,16)(19,20)(21,23)(25,26,31,32)(27,36,33,28)(29,34,30,35)(37,40,38,39);(1,3,5,8)(2,6,4,7)(9,10,15,18)(11,16,12,14)(13,17,22,24)(19,23,20,21)(26,32)(27,33)(34,35)(37,38);(9,11,15,12)(10,16,18,14)(13,19,22,20)(17,23,24,21)(25,27,31,33)(26,28,32,36)(29,37,30,38)(34,39,35,40);(9,12,15,11)(10,14,18,16)(13,19,22,20)(17,23,24,21)(25,28)(26,33)(27,32)(29,39)(30,40)(31,36)(34,38)(35,37);(1,2,5,4)(3,6,8,7)(9,13,15,22)(10,17,18,24)(11,19,12,20)(14,21,16,23)(25,29,31,30)(26,34,32,35)(27,37,33,38)(28,39,36,40);(1,4,5,2)(3,7,8,6)(9,14)(10,11)(12,18)(13,21)(15,16)(17,19)(20,24)(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,d^2=hi,e^2=ghi,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, 7554186);