#144088

order	512
nilpotency class	4
isoclinism	isoclinic class of #144085
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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

512 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,5)(4,7)(6,8)(9,10,17,24)(11,25,31,57)(12,32,37,67)(13,21,39,52)(14,20,38,53)(15,23,16,22)(18,41,46,27)(19,47,51,34)(26,58,63,70)(28,45,64,44)(29,56,30,43)(33,50,40,49)(35,54,36,48)(42,62,71,61)(55,68,72,60)(59,66,65,69);(2,5)(3,8)(6,7)(9,11,17,31)(10,18,24,46)(12,26,37,63)(13,28,39,64)(14,27,38,41)(15,30,16,29)(19,42,51,71)(20,43,53,56)(21,25,52,57)(22,45,23,44)(32,66,67,69)(33,68,40,60)(34,65,47,59)(35,62,36,61)(48,72,54,55)(49,70,50,58);(3,8)(5,6)(9,12)(10,19)(11,26)(13,34)(14,40)(15,36)(16,35)(17,37)(18,42)(20,32)(21,54)(22,50)(23,49)(24,51)(25,55)(27,60)(28,65)(29,62)(30,61)(31,63)(33,38)(39,47)(41,68)(43,66)(44,70)(45,58)(46,71)(48,52)(53,67)(56,69)(57,72)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51284);