#440906

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15670
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=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, 348102);