#251922

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,d^2=hi,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, 159118);