#133128

order	512
nilpotency class	4
isoclinism	isoclinic class of #133080
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#392	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 875 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40324);