#502379

order	512
nilpotency class	4
isoclinism	isoclinic class of #502377
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25321
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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)(3,16)(4,23,28,10)(5,29)(6,15)(7,11)(8,13)(9,19)(12,30)(14,34)(17,48,51,61)(18,44)(20,46)(21,36)(22,38)(24,57,58,63)(25,42,60,56)(26,41,27,47)(31,43)(32,37)(33,45,59,40)(35,54,62,50)(39,49,55,53)(52,64);(1,3,6,19,8,20,30,22)(2,9,12,16,13,38,15,46)(4,24,27,53,28,58,26,49)(5,18,7,32,31,21,34,52)(10,39,47,57,23,55,41,63)(11,36,29,37,14,44,43,64)(17,33,54,25,51,59,50,60)(35,40,61,42,62,45,48,56);(1,4)(2,10)(3,17)(5,25)(6,26)(7,33)(8,28)(9,35)(11,40)(12,41)(13,23)(14,45)(15,47)(16,48)(18,49)(19,50)(20,51)(21,53)(22,54)(24,52)(27,30)(29,56)(31,60)(32,58)(34,59)(36,39)(37,63)(38,62)(42,43)(44,55)(46,61)(57,64);(1,5,30,34,8,31,6,7)(2,11,15,43,13,14,12,29)(3,18,22,52,20,21,19,32)(4,25,27,59,28,60,26,33)(9,36,46,64,38,44,16,37)(10,40,47,42,23,45,41,56)(17,49,54,24,51,53,50,58)(35,39,61,57,62,55,48,63)

Polycyclic presentation:

2^9|a^2=efhi,b^2=fh,d^2=ei,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=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, 409575);