#7658052

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

4473 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)(10,16)(12,19)(17,20)(21,23)(25,26,31,35)(27,33,28,32)(29,30,34,40)(36,39,38,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,20)(12,18)(13,17)(14,19)(15,16)(21,22)(23,24)(25,27,31,28)(26,32,35,33)(29,36,34,38)(30,37,40,39);(1,3,4,7)(2,5,6,8)(9,11,15,13)(10,17,16,20)(12,21,19,23)(14,22,18,24)(25,28,31,27)(26,33,35,32)(29,38,34,36)(30,39,40,37);(9,12,15,19)(10,18,16,14)(11,21,13,23)(17,24,20,22);(9,13,15,11)(10,17,16,20)(12,23,19,21)(14,22,18,24)(25,29)(26,30)(27,36)(28,38)(31,34)(32,37)(33,39)(35,40);(9,14,15,18)(10,19,16,12)(11,22,13,24)(17,23,20,21)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,38)(33,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=i,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, 7565248);