#139494

order	512
nilpotency class	4
isoclinism	isoclinic class of #139432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

679 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 679 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=gh,d^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,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267641
maximal normal subgroup of	#72267641
maximal quotient of	#72267646

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46690);