#7657253

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

4534 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)(3,5,7,8)(12,20)(17,22)(19,21)(23,24)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,39,37);(1,2)(3,5)(4,6)(7,8)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(25,27,31,38)(26,32,35,36)(28,29,34,39)(30,37,40,33);(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,27,31,38)(26,32,35,36)(28,29,34,39)(30,37,40,33);(1,3,4,7)(2,5,6,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,28,31,34)(26,30,35,40)(27,29,38,39)(32,37,36,33);(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,29)(26,33)(27,34)(28,38)(30,32)(31,39)(35,37)(36,40);(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,30)(26,34)(27,37)(28,35)(29,36)(31,40)(32,39)(33,38)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=g,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, 7564449);