#8424533

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4304 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8331729);