#576715

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 483911);