#500260

order	512
nilpotency class	4
isoclinism	isoclinic class of #500256
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	#3487	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

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

normal subgroups only

This group has 816 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)(3,14,15,39)(4,19,20,34)(5,8,24,11)(6,12,28,40)(9,27,33,18)(10,25,37,22)(16,44,21,54)(17,47,43,41)(23,52,53,61)(26,30,50,38)(29,58,32,49)(31,60,35,62)(36,57,64,56)(42,48,63,45)(46,59,55,51);(2,8)(3,15)(4,20)(5,24)(9,29)(10,34)(11,13)(12,30)(14,41)(19,37)(23,53)(26,50)(31,51)(32,33)(35,59)(36,61)(38,40)(39,47)(42,44)(45,48)(46,55)(49,58)(52,64)(54,63);(1,3,5,17,7,15,24,43)(2,9,11,32,13,33,8,29)(4,21,22,48,20,16,25,45)(6,27,26,58,28,18,50,49)(10,35,19,51,37,31,34,59)(12,39,38,47,40,14,30,41)(23,46,56,62,53,55,57,60)(36,42,52,44,64,63,61,54);(1,4,7,20)(2,10,13,37)(3,16,15,21)(5,25,24,22)(6,23,28,53)(8,19,11,34)(9,31,33,35)(12,36,40,64)(14,42,39,63)(17,48,43,45)(18,46,27,55)(26,57,50,56)(29,59,32,51)(30,52,38,61)(41,54,47,44)(49,60,58,62)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,e^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,d^c=dh,f^c=fi,g^c=gi,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, 407456);