#146072

order	512
nilpotency class	4
isoclinism	isoclinic class of #146042
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

241 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=egh,c^2=d,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=eg,f^c=fh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53268);