#512972

order	512
nilpotency class	5
isoclinism	isoclinic class of #512972
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 336 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)(3,15,18,49)(4,7)(5,21)(6,25)(10,32,33,59)(11,14)(12,38)(16,50,28,46)(17,51,52,54)(19,53)(20,27)(22,26,48,47)(23,55)(24,45)(29,34,44,60)(30,40)(31,43)(35,39,57,64)(36,58)(41,62)(42,61,63,56);(1,2,4,9,8,11,7,14)(3,10,16,29,18,33,28,44)(5,12,19,30,24,36,23,41)(6,13,20,31,27,37,25,43)(15,34,46,32,49,60,50,59)(17,35,26,42,52,57,47,63)(21,40,55,38,45,62,53,58)(22,39,54,56,48,64,51,61);(1,3)(2,10)(4,16)(5,22)(6,26)(7,28)(8,18)(9,29)(11,33)(12,39)(13,42)(14,44)(15,45)(17,25)(19,54)(20,52)(21,49)(23,51)(24,48)(27,47)(30,56)(31,57)(32,58)(34,62)(35,43)(36,64)(37,63)(38,59)(40,60)(41,61)(46,53)(50,55);(2,11)(9,14)(10,33)(12,36)(13,37)(29,44)(30,41)(31,43)(32,59)(34,60)(35,57)(38,58)(39,64)(40,62)(42,63)(56,61)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,e^2=i,f^2=ghi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^c=fgi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224992, #72224998, #72224999, …
maximal normal subgroup of	#72224992, #72224998, #72224999, …
maximal quotient of	C₂×#512972

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420168);