#449975

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15954
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

987 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357171);