#251906

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:

(1,2,7,13)(3,8,16,28)(4,9,21,34)(5,10,24,38)(6,12,25,11)(14,30,41,26)(15,27,45,61)(17,29,46,39)(18,47,49,58)(19,50,22,53)(20,32,52,33)(23,36,56,37)(31,55,35,51)(40,63,42,64)(43,60,44,62)(48,59,54,57);(1,3,6,17)(2,8,12,29)(4,18)(5,15,23,44)(7,16,25,46)(9,30)(10,27,36,62)(11,39,13,28)(14,20)(19,48)(21,49)(22,54)(24,45,56,43)(26,34)(31,59)(32,47)(33,58)(35,57)(37,60,38,61)(40,51)(41,52)(42,55)(50,63)(53,64);(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=i,b^2=eh,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, 159102);