#8095923

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003119);