#429674

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15406
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

875 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336870);