#142925

order	512
nilpotency class	4
isoclinism	isoclinic class of #142828
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18,22,26)(19,27,30,46)(20,32,35,45)(21,36,34,25)(23,38,40,29)(24,42,43,33)(28,44,47,41)(31,48,37,39);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19,22,30)(18,23,26,40)(20,28,35,47)(21,37,34,31)(24,39,43,48)(25,44,36,41)(27,45,46,32)(29,42,38,33);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(19,28)(20,33)(21,34)(23,39)(24,32)(25,36)(27,41)(29,31)(30,47)(35,42)(37,38)(40,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,h^b=hi,d^c=dh,f^c=fg,g^c=gi,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, 50121);