#350450

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23930	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1366 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1366 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,d^b=dhi,f^c=fh,g^c=gi,f^d=fi,g^d=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, 257646);