#486699

order	512
nilpotency class	4
isoclinism	isoclinic class of #486696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18203
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

661 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 661 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gi,e^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,h^a=hi,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307081, #72307084, #72307112, …
maximal normal subgroup of	#72307081, #72307084, #72307112, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393895);