#561444

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 468640);