#8943436

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

4685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=g,e^2=g,f^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, 8850632);