#512039

order	512
nilpotency class	4
isoclinism	isoclinic class of #511840
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250202, #72250212, #72250223, …
maximal normal subgroup of	#72250202, #72250212, #72250223, …
maximal quotient of	#72257195

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419235);