#194372

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1276 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)(10,14,16,20)(11,17,19,18);(2,6)(3,4)(7,8)(9,10,12,20,13,16,21,14)(11,15,18,23,19,24,17,22);(9,11)(10,15)(12,18)(13,19)(14,22)(16,24)(17,21)(20,23);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,14,16,20)(11,18,19,17)(15,22,24,23)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101568);