#8445731

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

4293 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8352927);