#496393

order	512
nilpotency class	4
isoclinism	isoclinic class of #480674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1376 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1376 conjugacy classes of subgroups — too many to draw. Only its 210 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)(3,6,8,5)(4,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(2,5)(3,8)(6,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23);(10,16)(14,17)(15,24)(22,23)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403589);