#557667

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4518 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=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, 464863);