#7642163

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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:

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

Polycyclic presentation:

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

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, 7549359);