#8398978

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

4906 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,e^2=h,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, 8306174);