#8120078

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

4596 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8027274);