#8131525

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

4396 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)(10,13)(11,17)(16,21)(18,23)(25,26,31,35)(27,32,36,40)(28,30,38,33)(29,39,37,34);(1,2,5,3)(4,6,8,7)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,27,31,36)(26,32,35,40)(28,37,38,29)(30,34,33,39);(2,3)(6,7)(11,17)(12,20)(15,19)(16,21)(25,28,31,38)(26,33,35,30)(27,29,36,37)(32,34,40,39);(1,2,5,3)(4,6,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28)(26,30)(27,29)(31,38)(32,39)(33,35)(34,40)(36,37);(1,3,5,2)(4,7,8,6)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,34)(27,28)(30,40)(31,37)(32,33)(35,39)(36,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,19,17,15)(12,21,20,16)(18,24,23,22)(25,30)(26,28)(27,34)(29,40)(31,33)(32,37)(35,38)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8038721);