#8402377

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

4509 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309573);