#8757773

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

4921 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8664969);