#8230746

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

4973 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)(17,18,24,28)(19,29,31,25)(20,21,22,26)(23,27,30,32);(1,2)(3,4)(5,6)(7,8)(11,15)(14,16)(17,19)(18,25)(20,27)(21,23)(22,32)(24,31)(26,30)(28,29);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,20,24,22)(18,26,28,21)(19,27,31,32)(23,25,30,29);(9,11)(10,14)(12,15)(13,16)(17,21)(18,20)(19,30)(22,28)(23,31)(24,26)(25,32)(27,29);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,22,24,20)(18,21,28,26)(19,27,31,32)(23,25,30,29);(1,4,5,7)(2,3,6,8)(17,23,24,30)(18,27,28,32)(19,21,31,26)(20,29,22,25)

Polycyclic presentation:

2^9|a^2=g,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=dh,e^b=egi,f^c=fgi

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, 8137942);