#151765

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₁₆
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂³	C₂×D₁₆
prime radical	C₂⁴	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 218 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)(3,4)(5,6,7,8)(9,10)(11,24)(12,19)(13,21)(14,20)(15,39)(16,41)(17,23)(18,26)(22,34)(25,52)(27,53)(28,63)(29,65)(30,54)(31,47)(32,46)(33,51)(35,66)(36,50)(37,49)(38,48)(40,67)(42,55)(43,58)(44,61)(45,60)(56,70)(57,69)(59,68)(62,71)(64,72);(2,4)(9,11,14,27,15,28,31,55,12,25,32,56,33,57,36,60,17,30,38,62,40,64,35,59,16,29,37,61,34,58,13,26)(10,18,21,43,22,44,49,65,41,68,66,72,67,71,48,54,23,45,50,69,51,70,46,52,19,42,47,63,39,53,20,24);(9,12,17,16)(10,19,23,41)(11,25,30,29)(13,31,36,35)(14,32,38,37)(15,33,40,34)(18,42,45,68)(20,46,48,49)(21,47,50,66)(22,39,51,67)(24,52,54,65)(26,55,60,59)(27,56,62,61)(28,57,64,58)(43,63,69,72)(44,53,70,71)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,h^a=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, 58961);