#395472

order	512
nilpotency class	3
isoclinism	isoclinic class of #395471
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14724
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3369 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)(3,15,18,9)(4,21,24,10)(5,26,27,40)(6,13)(7,14)(8,12)(11,28,38,30)(16,41)(17,45)(19,34,20,33)(22,50)(23,53)(25,39,29,37)(31,44)(32,47)(35,48)(36,55)(42,49)(43,60,62,64)(46,59)(51,54)(52,61,63,58)(56,57);(1,3)(2,9)(4,22)(5,17)(6,20)(7,19)(8,18)(10,35)(11,32)(12,15)(13,34)(14,33)(16,25)(21,50)(23,52)(24,48)(26,45)(27,46)(28,49)(29,44)(30,47)(31,37)(36,60)(38,42)(39,41)(40,59)(43,54)(51,64)(53,58)(55,62)(56,63)(57,61);(1,4)(2,10)(3,16)(5,23)(6,29)(7,25)(8,24)(9,31)(11,36)(12,21)(13,39)(14,37)(15,41)(17,43)(18,44)(19,48)(20,22)(26,57)(27,56)(28,55)(30,54)(32,58)(33,50)(34,35)(38,51)(40,53)(42,61)(45,60)(46,62)(47,63)(49,52)(59,64);(1,5,8,27)(2,11,12,38)(3,17,18,46)(4,23,24,56)(6,28,7,30)(9,32,15,42)(10,36,21,51)(13,26,14,40)(16,43,44,62)(19,47,20,49)(22,52,48,63)(25,54,29,55)(31,58,41,61)(33,59,34,45)(35,60,50,64)(37,53,39,57)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302668);