#129719

order	512
nilpotency class	4
isoclinism	isoclinic class of #124838
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1507
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8328
prime radical	#60444	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

936 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,12,14,28)(4,17,19,32)(5,10)(6,22)(7,11)(8,13)(9,20,26,21)(15,25,16,31)(18,27)(23,29)(24,30);(1,3)(2,8)(4,18)(5,16)(6,15)(7,14)(9,25)(10,27)(11,23)(12,17)(13,21)(19,30)(20,29)(22,24)(26,31)(28,32);(4,19)(9,26)(12,28)(13,29)(17,32)(18,30)(20,21)(25,31)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,h^b=hi,d^c=di,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219613, #72224554, #72224559
maximal normal subgroup of	#72219613, #72224554, #72224559

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36915);