#460671

order	512
nilpotency class	3
isoclinism	isoclinic class of #460670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16369
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367867);