#251890

order	512
nilpotency class	3
isoclinism	isoclinic class of #251883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

798 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159086);