#137453

order	512
nilpotency class	4
isoclinism	isoclinic class of #12301
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9045
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,13,23,15,6,12,8,16,25,11,7,14,24,32)(3,10,18,28,21,26,22,29,20,31,17,27,19,30,4,9);(1,3,7,21,8,20,23,19)(2,9,14,28,16,29,15,27)(4,6,18,5,22,24,17,25)(10,13,26,32,31,11,30,12);(1,4)(2,10)(3,6)(5,21)(7,18)(8,22)(9,13)(11,29)(12,27)(14,26)(15,30)(16,31)(17,23)(19,25)(20,24)(28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gi,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egh,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fhi,g^c=gi,h^c=hi,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286542
maximal normal subgroup of	#72286542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44649);