#430119

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15413
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

927 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:

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

Polycyclic presentation:

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

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, 337315);