#139319

order	512
nilpotency class	4
isoclinism	isoclinic class of #139228
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,e^2=gh,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,e^c=egh,f^d=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, 46515);