#433501

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

741 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:

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

Polycyclic presentation:

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

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, 340697);