#133968

order	512
nilpotency class	4
isoclinism	isoclinic class of #133885
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

909 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 909 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233270, #72261275, #72261309
maximal normal subgroup of	#72233270, #72261275, #72261309
maximal quotient of	#72233268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41164);