#198268

order	512
nilpotency class	3
isoclinism	isoclinic class of #198117
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	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1330 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105464);