#153019

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

232 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60215);