#137445

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
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

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

normal subgroups only

This group has 374 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=g,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,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, 44641);