#8614115

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

4519 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=gi,f^2=gh,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, 8521311);