#133135

order	512
nilpotency class	4
isoclinism	isoclinic class of #133093
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

928 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,f^b=fghi,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, 40331);