#478247

order	512
nilpotency class	3
isoclinism	isoclinic class of #476143
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2819
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17137
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 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)(3,8,15,26)(4,9)(5,20,21,50)(6,12)(7,11)(10,22,33,51)(13,35,37,24)(14,39)(16,27,43,55)(17,44,45,49)(18,29)(19,30)(23,34)(25,54)(28,46,32,58)(31,42)(36,60)(38,59,62,53)(40,63)(41,48)(47,56)(52,61)(57,64);(1,3)(2,8,11,26)(4,13,18,37)(5,14)(6,16)(7,15)(9,24)(10,31,33,25)(12,27,34,55)(17,40,45,36)(19,38,47,62)(20,48,50,39)(21,41)(22,42)(23,43)(28,52)(29,35)(30,53)(32,57)(44,63)(46,64,58,61)(49,60)(51,54)(56,59);(1,4)(2,9)(3,13,15,37)(5,17,21,45)(6,19)(7,18)(8,24,26,35)(10,32,33,28)(11,29)(12,30)(14,40)(16,38,43,62)(20,49,50,44)(22,46,51,58)(23,47)(25,52)(27,53,55,59)(31,57)(34,56)(36,41)(39,63)(42,64)(48,60)(54,61);(1,5)(2,10,11,33)(3,14)(4,17,18,45)(6,22)(7,21)(8,25,26,31)(9,28)(12,20,34,50)(13,36,37,40)(15,41)(16,42)(19,46,47,58)(23,51)(24,52)(27,39,55,48)(29,32)(30,44)(35,57)(38,61,62,64)(43,54)(49,56)(53,63)(59,60)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^c=gi,e^d=ei,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, 385443);