#133265

order	512
nilpotency class	4
isoclinism	isoclinic class of #133265
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	#60808	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

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

normal subgroups only

This group has 618 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:

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

Polycyclic presentation:

2^9|a^2=f,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,e^b=ei,f^b=fghi,g^b=gi,d^c=di,f^c=fh,f^d=fi,f^e=fi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286866, #72292717, C₂×#133265
maximal normal subgroup of	#72286866, #72292717, C₂×#133265
maximal quotient of	#72286556, C₂×#133265, #72292835

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40461);