#151636

order	512
nilpotency class	5
isoclinism	isoclinic class of #151634
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12424
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#13692	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

159 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=dfi,d^2=fi,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fi,g^a=gh,h^a=hi,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225339, #72225565
maximal normal subgroup of	#72225339, #72225565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58832);