#7646337

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

4547 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=g,e^2=i,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, 7553533);