#134986

order	512
nilpotency class	4
isoclinism	isoclinic class of #134985
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

288 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5)(3,7)(6,8)(9,10)(11,21,24,16)(12,26)(13,30)(14,18)(15,36)(17,28)(19,35)(20,33)(22,49,50,68)(23,53,31,64)(25,40,56,48)(27,60)(29,63)(32,46)(34,59)(37,45)(38,51,47,62)(39,57,44,66)(41,61)(42,67)(43,54)(52,70,58,72)(55,69,65,71);(1,2,3,5,4,6,7,8)(9,11,15,25,14,24,33,56)(10,16,36,48,18,21,20,40)(12,22,37,58,27,50,29,52)(13,31,35,66,34,23,54,57)(17,38,42,69,41,47,46,71)(19,44,59,64,43,39,30,53)(26,49,45,72,60,68,63,70)(28,62,67,55,61,51,32,65);(9,12,14,27)(10,17,18,41)(11,22,24,50)(13,32,34,67)(15,37,33,29)(16,38,21,47)(19,45,43,63)(20,46,36,42)(23,55,31,65)(25,58,56,52)(26,59,60,30)(28,54,61,35)(39,70,44,72)(40,71,48,69)(49,64,68,53)(51,66,62,57)

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=i,d^2=gi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42182);