#8595739

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

4619 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8502935);