#145770

order	512
nilpotency class	4
isoclinism	isoclinic class of #145736
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,7)(6,8)(9,10,14,18)(11,16,20,24)(12,21,19,15)(13,23,22,17);(3,7)(9,11,14,20)(10,15,18,21)(12,22,19,13)(16,17,24,23);(1,2)(3,8)(4,6)(5,7)(9,12,13,11,14,19,22,20)(10,16,17,21,18,24,23,15)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=d,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fhi,c^b=cf,d^b=dg,e^b=eh,f^b=fg,e^c=ei,h^c=hi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52966);