#138527

order	512
nilpotency class	4
isoclinism	isoclinic class of #138491
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	D₈
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

485 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 82 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)(17,18,21,26,20,32,33,43,23,27,36,44,22,25,35,41)(19,28,30,38,37,48,46,40,31,45,47,39,29,42,34,24);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19,23,31)(18,24,27,40)(20,29,22,37)(21,34,36,46)(25,38,32,39)(26,42,44,48)(28,43,45,41)(30,33,47,35);(2,8)(4,5)(7,15)(11,13)(17,20,23,22)(18,25,27,32)(19,29,31,37)(21,35,36,33)(24,38,40,39)(26,43,44,41)(28,42,45,48)(30,46,47,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=i,d^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,f^c=fgi

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, 45723);