#151995

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	D₁₆
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₄×D₁₆
prime radical	C₂³	#508
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 202 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=deg,d^2=gh,e^2=ghi,g^2=hi,h^2=i,b^a=bd,d^b=dg,e^b=eg,g^b=gh,h^b=hi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59191);