#134972

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

292 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 106 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)(7,19)(8,25,27,40)(9,30)(10,15)(11,35)(12,18)(13,21)(14,26,41,61)(16,32)(17,31)(20,50,52,64)(22,47)(23,37)(24,49)(28,60,29,45)(33,42,62,43)(34,44)(36,54,63,53)(38,58)(39,56)(46,51,59,67)(48,55)(57,66,68,65);(5,7,10,22,11,23,32,56,12,24,34,58,31,55,9,21)(6,13,30,48,17,38,44,49,18,39,16,37,35,47,15,19)(8,20,33,57,28,53,61,67,27,52,62,68,29,54,26,51)(14,36,60,66,42,50,25,59,41,63,45,65,43,64,40,46);(5,8)(6,14)(7,20)(9,26)(10,33)(11,28)(12,27)(13,36)(15,40)(16,45)(17,42)(18,41)(19,46)(21,51)(22,57)(23,53)(24,52)(25,44)(29,31)(30,60)(32,61)(34,62)(35,43)(37,65)(38,50)(39,63)(47,64)(48,66)(49,59)(54,55)(56,67)(58,68)

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^c=ehi,f^c=fi,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, 42168);