#536618

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

3050 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 443814);