#200055

order	512
nilpotency class	3
isoclinism	isoclinic class of #199806
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1218 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107251);