#273699

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

902 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fh,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180895);