#455393

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16263
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 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)(3,13,16,8)(4,9,20,34)(5,22)(6,25)(7,11)(10,24)(12,28)(14,29)(15,30,44,41)(17,31,26,38)(18,47)(19,33,49,36)(21,39,27,35)(23,37)(32,46)(40,43)(42,55)(45,62,52,57)(48,60)(50,61,53,59)(51,54)(56,63)(58,64);(1,3)(2,8)(4,18)(5,15)(6,26)(7,16)(9,32)(10,30)(11,13)(12,38)(14,21)(17,28)(19,48)(20,46)(22,41)(23,52)(24,44)(25,31)(27,43)(29,35)(33,58)(34,47)(36,60)(37,62)(39,40)(42,50)(45,54)(49,64)(51,57)(53,63)(55,59)(56,61);(1,4)(2,9,11,34)(3,14)(5,19)(6,27)(7,20)(8,29,13,40)(10,36,22,33)(12,39,25,35)(15,42)(16,43)(17,46)(18,26)(21,28)(23,53)(24,49)(30,56,41,55)(31,47,38,32)(37,59,51,61)(44,63)(45,64)(48,52)(50,54)(57,58,62,60);(1,5)(2,10,11,22)(3,15)(4,19)(6,23)(7,24)(8,30,13,41)(9,33,34,36)(12,37,25,51)(14,42)(16,44)(17,45)(18,48)(20,49)(21,50)(26,52)(27,53)(28,54)(29,55,40,56)(31,57,38,62)(32,58,47,60)(35,59,39,61)(43,63)(46,64)

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 362589);