#289146

order	512
nilpotency class	3
isoclinism	isoclinic class of #289115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28472	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,g^b=gh,d^c=dh,f^c=fi,g^c=ghi

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, 196342);