#500272

order	512
nilpotency class	4
isoclinism	isoclinic class of #500268
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24058
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3500	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

817 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,e^b=ei,f^b=fh,g^b=gi,d^c=dh,f^c=fi,h^c=hi,e^d=ei,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407468);