#407265

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14955
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

961 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314461);