#289991

order	512
nilpotency class	3
isoclinism	isoclinic class of #289947
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	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=hi,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,g^b=gh,f^c=fi,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, 197187);