#288644

order	512
nilpotency class	3
isoclinism	isoclinic class of #288643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

739 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=eh,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195840);