#557700

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,e^2=f,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, 464896);