#145444

order	512
nilpotency class	4
isoclinism	isoclinic class of #144073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

258 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=dh,d^2=g,e^2=hi,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dgi,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=di,e^c=eghi,f^c=fi,h^c=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, 52640);