#137459

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1879
prime radical	#33660	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

333 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44655);