#253398

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

716 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=hi,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, 160594);