#153083

order	512
nilpotency class	5
isoclinism	isoclinic class of #153083
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

365 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|d^2=gh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60279);