#237999

order	512
nilpotency class	3
isoclinism	isoclinic class of #237997
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
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

986 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145195);