#144876

order	512
nilpotency class	4
isoclinism	isoclinic class of #142258
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 414 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,g^b=gi,h^b=hi,e^c=eg,f^c=fg,g^c=gi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52072);