#142704

order	512
nilpotency class	4
isoclinism	isoclinic class of #142624
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#33752	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 449 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49900);