#486078

order	512
nilpotency class	4
isoclinism	isoclinic class of #486077
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18178
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#23794	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 733 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281886, #72307468
maximal normal subgroup of	#72281886, #72307468

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393274);