#8617973

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

4542 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,e^2=gi,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, 8525169);