#512042

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

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

normal subgroups only

This group has 739 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)(3,16,18,9,21,33,19,35)(4,22,23,10,25,41,27,38)(7,11,26,40,30,14,28,39)(17,45,46,42,48,31,50,60)(20,43,49,32,53,44,51,34)(24,54,56,36,29,55,57,37)(47,59,63,61,52,58,64,62);(1,3,5,18,8,21,6,19)(2,9,12,16,15,35,13,33)(4,17,23,46,25,48,27,50)(7,20,26,49,30,53,28,51)(10,31,22,42,38,45,41,60)(11,32,39,43,14,34,40,44)(24,47,56,63,29,52,57,64)(36,58,54,61,37,59,55,62);(1,4)(2,10)(3,17)(5,23)(6,27)(7,29)(8,25)(9,31)(11,36)(12,22)(13,41)(14,37)(15,38)(16,42)(18,46)(19,50)(20,52)(21,48)(24,30)(26,57)(28,56)(32,58)(33,60)(34,59)(35,45)(39,54)(40,55)(43,61)(44,62)(47,53)(49,64)(51,63);(2,11)(7,30)(9,32)(10,36)(12,39)(13,40)(14,15)(16,43)(20,53)(22,54)(24,29)(26,28)(31,58)(33,44)(34,35)(37,38)(41,55)(42,61)(45,59)(47,52)(49,51)(56,57)(60,62)(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,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419238);