#8935428

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

4730 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842624);