#512044

order	512
nilpotency class	4
isoclinism	isoclinic class of #511841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#32976
prime radical	#3529	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

737 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ci,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250047, #72250052, #72250212, …
maximal normal subgroup of	#72250047, #72250052, #72250212, …
maximal quotient of	#72250465, #72279633

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419240);