#251046

order	512
nilpotency class	3
isoclinism	isoclinic class of #251031
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

1129 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=hi,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,d^b=dh,e^b=eh,g^b=gi,f^c=fi,g^c=gi,e^d=eh

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, 158242);