#144136

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33716	C₂
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

499 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,6)(4,7)(5,8)(9,10)(11,23)(12,18)(13,34)(14,19)(15,38)(16,22)(17,25)(20,36)(21,35)(24,45)(26,53)(27,55)(28,43)(29,64)(30,51)(31,46)(32,56)(33,49)(37,52)(39,50)(40,48)(41,65)(42,61)(44,62)(47,67)(54,68)(57,59)(58,72)(60,71)(63,69)(66,70);(2,6)(3,4)(7,8)(9,11)(10,17)(12,29)(13,27)(14,25)(15,39)(16,26)(18,45)(19,50)(20,53)(21,55)(22,42)(23,34)(24,31)(28,35)(30,60)(32,58)(33,65)(36,62)(37,61)(38,44)(40,66)(41,47)(43,52)(46,72)(48,71)(49,57)(51,64)(54,70)(56,69)(59,68)(63,67);(2,7)(3,4)(6,8)(9,12,16,33)(10,18,22,49)(11,24,26,59)(13,31,36,68)(14,30,37,67)(15,40,35,32)(17,41,42,64)(19,51,52,47)(20,54,34,46)(21,56,38,48)(23,57,53,45)(25,29,61,65)(27,58,62,66)(28,63,39,60)(43,71,50,69)(44,72,55,70)

Polycyclic presentation:

2^9|c^2=i,d^2=gi,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fghi,g^b=gi,d^c=dg,e^c=eh,f^c=fghi,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, 51332);