#146262

order	512
nilpotency class	4
isoclinism	isoclinic class of #146219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 257 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:

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

Polycyclic presentation:

2^9|a^2=dgh,b^2=de,c^2=i,d^2=g,e^2=g,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,d^c=dgi,e^c=egi,f^c=fhi,g^c=gi

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, 53458);