#469830

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16664
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

848 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)(3,14)(4,10)(5,24)(6,29)(7,13)(8,12)(9,17)(11,27)(15,43)(16,46)(18,33)(19,42)(20,31)(21,35)(22,37)(23,38)(25,54)(26,40)(28,55)(30,36)(32,49)(34,45)(39,57)(41,50)(44,60)(47,58)(48,53)(51,59)(52,62)(56,61)(63,64);(1,3,6,17,7,19,8,18)(2,9,12,14,13,33,29,42)(4,20,22,45,23,51,30,15)(5,16,26,48,27,50,28,49)(10,34,36,31,38,43,37,59)(11,32,40,41,24,53,55,46)(21,47,25,56,52,64,57,44)(35,61,39,58,62,60,54,63);(1,4,8,22,7,23,6,30)(2,10,29,36,13,38,12,37)(3,15,17,51,19,45,18,20)(5,25,28,52,27,57,26,21)(9,31,14,34,33,59,42,43)(11,39,55,62,24,54,40,35)(16,47,48,44,50,64,49,56)(32,60,41,58,53,61,46,63);(1,5)(2,11,13,24)(3,16)(4,21,23,52)(6,26)(7,27)(8,28)(9,32,33,53)(10,35)(12,40,29,55)(14,41,42,46)(15,44,45,56)(17,48)(18,49)(19,50)(20,47,51,64)(22,25,30,57)(31,58)(34,61)(36,39)(37,54)(38,62)(43,60)(59,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377026);