#145056

order	512
nilpotency class	4
isoclinism	isoclinic class of #143329
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1634
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10412
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

281 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 281 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=eg,f^c=fhi,h^c=hi

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, 52252);