#135769

order	512
nilpotency class	4
isoclinism	isoclinic class of #135766
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1846
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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)(11,21,24,42)(12,26,28,17)(13,30)(14,33)(15,19)(16,23,36,48)(18,31)(20,34)(22,44)(25,51,50,43)(27,55,56,59)(29,39,58,54)(32,60)(35,45)(37,64,62,49)(38,57,53,71)(40,61)(41,47)(46,65)(52,69)(63,72)(66,68)(67,70);(9,11,13,23,14,25,31,49,15,24,32,48,34,50,61,64)(10,16,18,21,20,37,30,43,19,36,40,42,33,62,60,51)(12,22,27,45,29,46,57,68,28,47,56,69,58,70,71,72)(17,35,38,41,39,63,59,65,26,52,53,44,54,66,55,67);(1,2,3,5,4,6,7,8)(9,12,14,29,15,28,34,58)(10,17,33,54,19,26,20,39)(11,22,25,46,24,47,50,70)(13,27,31,57,32,56,61,71)(16,35,62,66,36,52,37,63)(18,38,60,55,40,53,30,59)(21,41,51,67,42,44,43,65)(23,45,49,68,48,69,64,72)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=eg,d^2=g,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42965);