#471960

order	512
nilpotency class	3
isoclinism	isoclinic class of #464193
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2657
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16809
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379156);