#7644341

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

4330 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)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38);(1,2)(3,5)(4,6)(7,8)(11,18)(15,17)(19,24)(22,23)(26,27)(31,33)(32,34)(38,39);(1,3,4,7)(2,5,6,8)(10,14)(15,17)(16,20)(22,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

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, 7551537);