#7746081

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	#43982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4356 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)(9,10,15,14)(11,19,20,16)(12,17,21,22)(13,23,24,18);(1,2)(3,7)(4,8)(5,6)(9,11,15,20)(10,16,14,19)(12,13,21,24)(17,18,22,23)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(10,14)(11,20)(12,21)(18,23)(26,30)(31,37)(32,35)(38,39);(9,12)(10,17)(11,13)(14,22)(15,21)(16,18)(19,23)(20,24)(25,27,29,34)(26,31,30,37)(28,33,36,40)(32,38,35,39);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,12)(14,23)(15,24)(16,17)(19,22)(20,21)(25,28)(26,32)(27,33)(29,36)(30,35)(31,38)(34,40)(37,39);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,19,20,16)(12,22,21,17)(13,23,24,18)(25,26)(27,31)(28,32)(29,30)(33,38)(34,37)(35,36)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7653277);