#133101

order	512
nilpotency class	4
isoclinism	isoclinic class of #133093
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

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

normal subgroups only

This group has 957 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,f^b=fgh,f^c=fhi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40297);