#474368

order	512
nilpotency class	3
isoclinism	isoclinic class of #467073
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2640
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16961
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

563 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 563 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=g,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,h^a=hi,c^b=cg,d^c=de,e^c=ei,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, 381564);