#479748

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17233
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

742 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)(3,15)(4,21,25,41)(5,26,29,11)(6,12)(7,13)(8,14)(9,18)(10,24,38,30)(16,45,47,36)(17,33,49,44)(19,35)(20,34)(22,32,51,43)(23,52)(27,39,50,57)(28,42,31,40)(37,54)(46,63)(48,61)(53,59)(55,58)(56,62)(60,64);(1,3)(2,9,14,15)(4,22,25,51)(5,27)(6,19)(7,20)(8,18)(10,36)(11,39,26,57)(12,34,13,35)(16,24,47,30)(17,28)(21,43)(23,48,56,46)(29,50)(31,49)(32,41)(33,42,44,40)(37,60)(38,45)(52,63)(53,55,64,54)(58,59)(61,62);(1,4)(2,10)(3,16,18,47)(5,23)(6,30)(7,24)(8,25)(9,32,15,43)(11,37)(12,41)(13,21)(14,38)(17,48,49,46)(19,51,20,22)(26,58)(27,53,50,64)(28,55)(29,56)(31,54)(33,60,44,59)(34,45,35,36)(39,61,57,63)(40,52)(42,62);(1,5)(2,11)(3,17)(4,23)(6,31)(7,28)(8,29)(9,33)(10,37)(12,42)(13,40)(14,26)(15,44)(16,46)(18,49)(19,27)(20,50)(21,52)(22,53)(24,55)(25,56)(30,54)(32,59)(34,39)(35,57)(36,61)(38,58)(41,62)(43,60)(45,63)(47,48)(51,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386944);