#8533383

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

4372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440579);