#513188

order	512
nilpotency class	5
isoclinism	isoclinic class of #513188
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #2170 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 118 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)(4,22)(5,27)(6,32)(7,29)(10,39)(11,23)(12,37)(13,47)(14,36)(15,38)(17,50)(18,25)(20,21)(24,55)(26,56)(28,33)(30,51)(31,59)(34,62)(35,40)(41,57)(42,64)(43,52)(44,58)(45,63)(46,61)(48,60)(49,53);(1,2)(3,10)(4,23)(5,28)(6,12)(7,33)(8,15)(9,27)(11,19)(13,29)(14,32)(16,40)(17,47)(18,52)(20,45)(21,42)(22,35)(24,57)(25,39)(26,58)(30,60)(31,61)(34,46)(36,50)(37,51)(38,59)(41,53)(43,49)(44,54)(48,62)(55,64)(56,63);(1,3)(2,10)(4,17)(5,18)(6,20)(7,19)(8,21)(9,35)(11,33)(12,45)(13,41)(14,40)(15,42)(16,32)(22,27)(23,47)(24,31)(25,51)(26,30)(28,52)(29,53)(34,54)(36,64)(37,39)(38,63)(43,48)(44,46)(49,62)(50,55)(56,59)(57,61)(58,60);(1,4)(2,11)(3,17)(5,16)(6,24)(7,22)(8,26)(9,35)(10,33)(12,43)(13,40)(14,41)(15,44)(18,32)(19,27)(20,31)(21,30)(23,28)(25,29)(34,56)(36,39)(37,64)(38,63)(42,46)(45,48)(47,52)(49,50)(51,53)(54,59)(55,62)(57,60)(58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225555, #72225561, #72225565
maximal normal subgroup of	#72225555, #72225561, #72225565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420384);