#488330

order	512
nilpotency class	4
isoclinism	isoclinic class of #487715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

740 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,f^e=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, 395526);