#8395066

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

6222 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=gi,f^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, 8302262);