#544453

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2480 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 451649);