#131079

order	512
nilpotency class	4
isoclinism	isoclinic class of #130736
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	#61409	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

622 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 128

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

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

Polycyclic presentation:

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

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, 38275);