#153021

order	512
nilpotency class	5
isoclinism	isoclinic class of #150157
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

234 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=eh,d^2=ghi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,h^a=hi,d^b=di,f^b=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, 60217);