#472664

order	512
nilpotency class	3
isoclinism	isoclinic class of #462392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2517
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16830
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379860);