#437195

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15586
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

804 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)(3,15,19,9)(4,20,25,41)(5,26)(6,13)(7,14)(8,12)(10,23,37,24)(11,27)(16,35)(17,46,49,55)(18,34,30,33)(21,42)(22,36,53,61)(28,39)(29,40)(31,44)(32,48,38,47)(43,57)(45,58)(50,54,56,52)(51,59)(60,63)(62,64);(1,3)(2,9)(4,21)(5,17)(6,18)(7,30)(8,19)(10,35)(11,38)(12,15)(13,33)(14,34)(16,23)(20,31)(22,51)(24,45)(25,44)(26,55)(27,47)(28,48)(29,49)(32,39)(36,60)(37,58)(40,46)(41,42)(43,52)(50,62)(53,63)(54,64)(56,57)(59,61);(1,4,6,23,8,25,7,24)(2,10,14,41,12,37,13,20)(3,16,18,44,19,45,30,21)(5,22,27,52,29,53,28,54)(9,31,34,35,15,42,33,58)(11,36,26,56,39,61,40,50)(17,43,47,63,49,64,48,51)(32,57,46,59,38,62,55,60);(1,5,6,27,8,29,7,28)(2,11,13,40,12,39,14,26)(3,17,18,47,19,49,30,48)(4,22,23,52,25,53,24,54)(9,32,33,55,15,38,34,46)(10,36,20,50,37,61,41,56)(16,43,44,63,45,64,21,51)(31,57,58,60,42,62,35,59)

Polycyclic presentation:

2^9|a^2=fg,c^2=ef,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=gi,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, 344391);