#253397

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	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 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:

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

Polycyclic presentation:

2^9|b^2=eh,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, 160593);