#8157062

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

4609 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,e^2=gh,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^d=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, 8064258);