#237949

order	512
nilpotency class	3
isoclinism	isoclinic class of #237948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1023 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,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, 145145);