#137428

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, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

421 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(2,4)(5,6,9,13,10,14,8,12)(7,15,17,11,18,19,16,20)(21,22,24,30,40,49,43,31,27,33,41,29,26,32,25,42)(23,34,35,51,44,45,36,48,38,50,52,47,37,46,39,28);(1,2,3,4)(5,7)(6,11)(8,17)(9,16)(10,18)(12,19)(13,15)(14,20)(21,23)(22,28)(24,39)(25,35)(26,44)(27,38)(29,45)(30,46)(31,50)(32,51)(33,48)(34,42)(36,41)(37,40)(43,52)(47,49);(5,7)(6,11)(8,16)(9,17)(10,18)(12,15)(13,19)(14,20)(22,29)(24,25)(26,40)(28,45)(30,33)(31,49)(32,42)(34,51)(35,39)(36,52)(37,44)(41,43)(46,48)(47,50)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gi,e^2=g,f^2=gi,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, 44624);