#317887

order	512
nilpotency class	3
isoclinism	isoclinic class of #317883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225083);