#418855

order	512
nilpotency class	3
isoclinism	isoclinic class of #418854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15266
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1891 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286134, #72286176, #72286235
maximal normal subgroup of	#72286134, #72286176, #72286235

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326051);