#138571

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

311 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 96 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)(5,6,10,16,32,14,43,36,9,18,19,15,11,17,35,44)(7,12,23,38,24,49,33,47,25,40,22,58,21,39,51,45)(8,26,29,50,30,48,34,42,31,62,28,41,27,46,52,13)(20,53,55,65,57,68,60,37,54,64,67,63,56,61,59,66);(5,7,19,51,32,21,35,22,9,25,10,33,11,24,43,23)(6,12,36,38,17,49,16,47,18,40,44,58,14,39,15,45)(8,20,52,67,27,57,28,59,31,54,34,55,30,56,29,60)(13,37,26,61,50,65,48,64,42,66,62,68,41,63,46,53);(2,4)(5,8)(6,13)(7,20)(9,31)(10,34)(11,30)(12,37)(14,41)(15,46)(16,48)(17,50)(18,42)(19,52)(21,57)(22,59)(23,60)(24,56)(25,54)(26,36)(27,32)(28,35)(29,43)(33,55)(38,61)(39,63)(40,66)(44,62)(45,53)(47,64)(49,65)(51,67)(58,68)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=ghi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,d^c=dh,e^c=eh,f^c=fhi,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45767);