#131356

order	512
nilpotency class	4
isoclinism	isoclinic class of #131354
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8403
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#60771	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

757 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339831
maximal normal subgroup of	#72339831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38552);