#512187

order	512
nilpotency class	4
isoclinism	isoclinic class of #511890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=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, 419383);