#451197

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15988
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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)(3,14)(4,19,23,10)(5,24)(6,13)(7,29)(8,12)(9,17)(11,27)(15,32,45,43)(16,47)(18,35)(20,36,46,50)(21,53,55,63)(22,39,30,38)(25,58)(26,41)(28,34)(31,42)(33,48)(37,57,51,56)(40,49)(44,62,59,61)(52,60,54,64);(1,3)(2,9,12,14)(4,20)(5,25,26,16)(6,28)(7,18)(8,17)(10,36,19,50)(11,40)(13,35,29,34)(15,22)(21,54,55,52)(23,46)(24,47)(27,48,31,49)(30,45)(32,39,43,38)(33,42)(37,62)(41,58)(44,56,59,57)(51,61)(53,60)(63,64);(1,4)(2,10)(3,15,17,45)(5,21)(6,22)(7,30)(8,23)(9,32,14,43)(11,37)(12,19)(13,38)(16,44,25,59)(18,20,28,46)(24,53)(26,55)(27,56)(29,39)(31,57)(33,60,40,64)(34,36,35,50)(41,63)(42,51)(47,62,58,61)(48,52,49,54);(1,5)(2,11)(3,16)(4,21)(6,27)(7,31)(8,26)(9,33)(10,37)(12,42)(13,41)(14,40)(15,44)(17,25)(18,48)(19,51)(20,52)(22,56)(23,55)(24,29)(28,49)(30,57)(32,60)(34,58)(35,47)(36,61)(38,63)(39,53)(43,64)(45,59)(46,54)(50,62)

Polycyclic presentation:

2^9|a^2=efgh,b^2=f,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,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, 358393);