#134649

order	512
nilpotency class	4
isoclinism	isoclinic class of #134475
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41845);