#448886

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15924
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 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)(3,15,20,34)(4,21,26,42)(5,27)(6,12)(7,13)(8,14)(9,18,35,19)(10,24,38,25)(11,29)(16,32)(17,47,49,33)(22,36)(23,37,54,61)(28,57,48,39)(30,41)(31,40)(43,45)(44,62)(46,50)(51,55,56,53)(52,59)(58,64)(60,63);(1,3,7,18,8,20,6,19)(2,9,13,15,14,35,12,34)(4,22,24,46,26,45,25,16)(5,28,30,49,31,48,29,17)(10,36,21,50,38,43,42,32)(11,39,27,47,41,57,40,33)(23,44,53,52,54,63,55,58)(37,59,51,60,61,64,56,62);(1,4,6,24,8,26,7,25)(2,10,12,21,14,38,13,42)(3,16,18,45,20,46,19,22)(5,23,29,53,31,54,30,55)(9,32,15,43,35,50,34,36)(11,37,40,51,41,61,27,56)(17,44,28,58,49,63,48,52)(33,59,39,62,47,64,57,60);(1,5,8,31)(2,11,14,41)(3,17)(4,23,26,54)(6,29,7,30)(9,33)(10,37,38,61)(12,40,13,27)(15,39)(16,44)(18,28)(19,48)(20,49)(21,51,42,56)(22,52)(24,53,25,55)(32,59)(34,57)(35,47)(36,60)(43,62)(45,58)(46,63)(50,64)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=eg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^c=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, 356082);