#8230865

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

4812 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=g,f^2=i,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, 8138061);