#391253

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

889 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)(3,14)(4,19,24,42)(5,11)(6,30)(7,12)(8,13)(9,17)(10,23,38,22)(15,36,20,32)(16,45)(18,34)(21,52,55,39)(25,56)(26,58,54,37)(27,43)(28,40)(29,41)(31,35)(33,51)(44,47,60,50)(46,63,53,59)(48,61,57,64)(49,62);(1,3,6,17,8,18,7,31)(2,9,12,14,13,35,30,34)(4,20,22,50,24,15,23,47)(5,25,27,51,29,16,28,49)(10,36,19,44,38,32,42,60)(11,33,40,56,41,62,43,45)(21,46,26,48,55,53,54,57)(37,63,39,64,58,59,52,61);(1,4)(2,10)(3,15)(5,26)(6,23)(7,22)(8,24)(9,32)(11,39)(12,42)(13,38)(14,44)(16,48)(17,50)(18,20)(19,30)(21,27)(25,57)(28,55)(29,54)(31,47)(33,61)(34,60)(35,36)(37,40)(41,52)(43,58)(45,63)(46,49)(51,53)(56,59)(62,64);(1,5,8,29)(2,11,13,41)(3,16,18,25)(4,21,24,55)(6,28,7,27)(9,33,35,62)(10,37,38,58)(12,43,30,40)(14,45,34,56)(15,46,20,53)(17,51,31,49)(19,52,42,39)(22,54,23,26)(32,59,36,63)(44,61,60,64)(47,57,50,48)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=i,e^2=i,f^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, 298449);