#8398561

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

5361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8305757);