#7718455

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

4141 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)(9,10,16,13)(11,20,22,17)(12,23,15,18)(14,24,21,19)(28,36)(31,38)(34,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,16,22)(10,17,13,20)(12,21,15,14)(18,19,23,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(9,12,16,15)(10,18,13,23)(11,21,22,14)(17,19,20,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,13,16,10)(11,20,22,17)(12,18,15,23)(14,24,21,19);(9,14)(10,19)(11,12)(13,24)(15,22)(16,21)(17,23)(18,20)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3,4,7)(2,5,6,8)(9,15,16,12)(10,18,13,23)(11,14,22,21)(17,19,20,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

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, 7625651);