#8387668

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

5376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,21)(22,24);(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,14)(13,16)(18,21)(22,23);(2,6)(4,7)(9,10,12,14)(11,13,16,15)(17,18)(19,22)(20,21)(23,24);(2,6)(3,8)(11,16)(13,15)(17,19,20,24)(18,22,21,23);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8294864);