#137373

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	#33516	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 566 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,4,6,5,3,8,7)(9,10,14,21,15,22,38,52,16,23,39,20,37,51,13,19)(11,24,27,46,28,43,26,55,29,53,36,17,40,44,64,45)(12,30,33,48,34,47,31,49,35,50,32,56,41,54,68,18)(25,58,61,67,62,70,65,72,63,71,60,69,66,42,59,57);(1,3)(2,5)(4,6)(7,8)(9,11,15,40,16,29,37,28)(10,17,22,55,23,46,51,45)(12,25,41,62,35,63,34,66)(13,36,14,26,38,27,39,64)(18,42,56,58,49,70,48,71)(19,43,21,24,52,44,20,53)(30,67,54,72,50,69,47,57)(31,61,33,65,68,60,32,59);(1,2)(3,5)(4,6)(7,8)(9,12)(10,18)(11,25)(13,32)(14,31)(15,41)(16,35)(17,42)(19,50)(20,54)(21,47)(22,56)(23,49)(24,57)(26,61)(27,65)(28,66)(29,63)(30,52)(33,38)(34,37)(36,59)(39,68)(40,62)(43,69)(44,67)(45,71)(46,70)(48,51)(53,72)(55,58)(60,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=g,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, 44569);