#237937

order	512
nilpotency class	3
isoclinism	isoclinic class of #10362
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#393
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,e^c=eh,f^c=fhi,g^c=gh

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, 145133);