#7658363

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

4322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565559);