#131097

order	512
nilpotency class	4
isoclinism	isoclinic class of #130730
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

540 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38293);