#134973

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,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, 42169);