#420928

order	512
nilpotency class	3
isoclinism	isoclinic class of #417617
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15282
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1487 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=i,f^2=i,g^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,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=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, 328124);