#138512

order	512
nilpotency class	4
isoclinism	isoclinic class of #138482
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

469 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fh,h^b=hi,d^c=dg,f^c=fg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45708);