#513190

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

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

normal subgroups only

This group has 422 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)(10,40)(11,23)(12,38)(13,49)(14,37)(15,39)(17,53)(18,25)(20,21)(24,57)(26,58)(28,33)(30,54)(31,35)(34,50)(36,41)(42,59)(43,64)(45,55)(46,60)(47,63)(48,62)(51,61)(52,56);(1,2,8,15)(3,10,21,43)(4,23,26,60)(5,28,31,62)(6,12,32,14)(7,33,34,48)(9,35,39,27)(11,44,46,19)(13,50,51,29)(16,41,20,47)(17,49,30,61)(18,55,24,59)(22,36,58,63)(25,40,57,64)(37,54,38,53)(42,52,45,56);(1,3)(2,10)(4,17)(5,18)(6,20)(7,19)(8,21)(9,36)(11,33)(12,47)(13,42)(14,41)(15,43)(16,32)(22,27)(23,49)(24,31)(25,54)(26,30)(28,55)(29,56)(34,44)(35,58)(37,64)(38,40)(39,63)(45,51)(46,48)(50,52)(53,57)(59,62)(60,61);(1,4)(2,11)(3,17)(5,16)(6,24)(7,22)(8,26)(9,36)(10,33)(12,45)(13,41)(14,42)(15,46)(18,32)(19,27)(20,31)(21,30)(23,28)(25,29)(34,58)(35,44)(37,40)(38,64)(39,63)(43,48)(47,51)(49,55)(50,57)(52,53)(54,56)(59,61)(60,62)

Polycyclic presentation:

2^9|b^2=i,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, #72225563, #72225565
maximal normal subgroup of	#72225555, #72225563, #72225565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420386);