#153315

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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)(5,6,13,21)(7,22,28,46)(8,14,33,43)(9,34,10,38)(11,39,36,42)(12,40,31,20)(15,23,49,25)(16,37,50,29)(17,32,18,35)(19,52,51,30)(24,59,64,44)(26,65,57,67)(27,53,61,41)(45,60,56,68)(47,63,55,66)(48,62,54,58);(6,14)(7,23)(8,29)(9,35)(10,32)(11,30)(12,31)(15,44)(16,19)(17,40)(18,20)(21,43)(22,53)(24,60)(25,28)(26,63)(27,62)(33,37)(34,39)(36,52)(38,42)(41,46)(47,48)(49,59)(50,51)(54,55)(57,66)(58,61)(64,68)(65,67);(2,4)(5,7,8,24,30,60,9,23,12,27,32,63,36,57,37,58,13,28,33,64,52,68,10,25,31,61,35,66,11,26,29,62)(6,15,16,45,39,44,17,46,20,48,38,67,51,55,43,53,21,49,50,56,42,59,18,22,40,54,34,65,19,47,14,41)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=gh,e^2=gi,f^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fghi,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, 60511);