#7656876

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

4616 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,e^2=gh,f^2=g,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, 7564072);