#432887

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15468
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,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, 340083);