#232707

order	512
nilpotency class	3
isoclinism	isoclinic class of #232706
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1346 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139903);