#450370

order	512
nilpotency class	3
isoclinism	isoclinic class of #418851
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2679
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15966
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1029 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^d=gi,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, 357566);