#470665

order	512
nilpotency class	3
isoclinism	isoclinic class of #464193
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2657
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16692
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

885 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:

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

Polycyclic presentation:

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

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, 377861);