#240976

order	512
nilpotency class	3
isoclinism	isoclinic class of #240963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

881 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,14)(13,17)(15,18)(16,21)(20,23)(22,24);(1,2)(3,7)(4,6)(5,8)(9,11,15,23)(10,16,18,24)(12,19,17,20)(13,21,14,22);(2,6)(3,4)(7,8)(9,12,14,10,15,17,13,18)(11,20,21,16,23,19,22,24);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,17,18,12)(11,21,23,22)(16,20,24,19)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=ghi,d^c=di,f^c=fhi,g^c=gi

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, 148172);