#536109

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2538 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)(11,16)(12,15)(13,14)(17,18,20,22)(19,21,23,24);(2,4)(5,7,8,6)(9,11,14,12)(10,15,13,16);(1,2,3,4)(11,12)(15,16)(17,18)(19,21)(20,22)(23,24);(9,12,14,11)(10,15,13,16)(17,18)(19,21)(20,22)(23,24);(9,13,14,10)(11,15,12,16)(17,19,20,23)(18,21,22,24)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fh,d^2=f,e^2=fg,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, 443305);