#244657

order	512
nilpotency class	3
isoclinism	isoclinic class of #244655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1192 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,22)(13,23)(14,18)(15,21)(17,20)(19,24);(1,2)(3,7)(4,6)(5,8)(9,11,14,19)(10,15,18,22)(12,20,21,13)(16,17,24,23);(2,6)(3,4)(7,8)(11,19)(13,20)(15,22)(17,23);(9,12)(10,16)(11,20)(13,19)(14,21)(15,17)(18,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151853);