#133629

order	512
nilpotency class	4
isoclinism	isoclinic class of #133406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator, prime radical	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

516 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,e^b=ehi,f^b=fg,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40825);