#8120062

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

4461 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)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,31,36)(27,29,34,32)(28,35,30,33)(37,40,38,39);(1,2,5,3)(4,6,8,7)(10,14)(15,17)(16,20)(22,23)(25,27)(26,32)(28,37)(29,36)(30,38)(31,34)(33,40)(35,39);(2,3)(6,7)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,27)(26,29)(28,37)(30,38)(31,34)(32,36)(33,39)(35,40);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,31,30)(26,33,36,35)(27,37,34,38)(29,39,32,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,29)(26,34)(27,36)(28,39)(30,40)(31,32)(33,38)(35,37);(1,4)(2,6)(3,7)(5,8)(9,12,13,21)(10,16,14,20)(11,18,19,24)(15,22,17,23)(25,30,31,28)(26,35,36,33)(27,38,34,37)(29,40,32,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 8027258);