#144968

order	512
nilpotency class	4
isoclinism	isoclinic class of #142220
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

487 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,e^b=eh,f^b=fg,e^c=eg,f^c=fg,h^c=hi,f^e=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, 52164);