#143036

order	512
nilpotency class	4
isoclinism	isoclinic class of #143009
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,e^c=egi,f^c=fg,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50232);