#253403

order	512
nilpotency class	3
isoclinism	isoclinic class of #253387
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, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh,f^c=fi,g^c=gh,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, 160599);