#144150

order	512
nilpotency class	4
isoclinism	isoclinic class of #144135
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
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

432 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=g,e^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,h^b=hi,d^c=dgi,e^c=eh,f^c=fgh,h^c=hi

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, 51346);