#391255

order	512
nilpotency class	3
isoclinism	isoclinic class of #391252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

878 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298451);