#512952

order	512
nilpotency class	5
isoclinism	isoclinic class of #512952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2215
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

423 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=eg,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gh,h^a=hi,d^b=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225555, #72225573, #72225574, …
maximal normal subgroup of	#72225555, #72225573, #72225574, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420148);