#134995

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,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, 42191);