#133262

order	512
nilpotency class	4
isoclinism	isoclinic class of #133259
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8522
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 624 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) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231251, #72231259, #72231264, …
maximal normal subgroup of	#72231251, #72231259, #72231264, …
maximal quotient of	#72279219, C₂×#133262

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40458);