#413818

order	512
nilpotency class	3
isoclinism	isoclinic class of #403928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2615
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15157
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1137 conjugacy classes of subgroups; only the 244 normal subgroups are shown.

normal subgroups only

This group has 1137 conjugacy classes of subgroups — too many to draw. Only its 244 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)(3,13)(4,18)(5,22)(6,25)(7,11)(8,16)(9,20)(10,23)(12,26)(14,30)(15,39)(17,29)(19,45)(21,32)(24,51)(27,37)(28,55)(31,58)(33,52)(34,41)(35,42)(36,59)(38,54)(40,49)(43,47)(44,48)(46,50)(53,64)(56,61)(57,62)(60,63);(1,3)(2,8,11,13)(4,14)(5,15)(6,17)(7,16)(9,30,18,27)(10,28,33,34)(12,29,25,35)(19,36)(20,37)(21,38)(22,49,51,39)(23,41)(24,40)(26,42)(31,56,43,53)(32,57,44,54)(45,63,50,59)(46,60)(47,61)(48,62)(52,55)(58,64);(1,4)(2,9,11,18)(3,14)(5,19)(6,21)(7,20)(8,27,13,30)(10,31,33,43)(12,32,25,44)(15,36)(16,37)(17,38)(22,50,51,45)(23,47)(24,46)(26,48)(28,53,34,56)(29,54,35,57)(39,59,49,63)(40,60)(41,61)(42,62)(52,58)(55,64);(1,5,6,23,7,24,26,52)(2,10,12,22,11,33,25,51)(3,15,17,41,16,40,42,55)(4,19,21,47,20,46,48,58)(8,28,29,49,13,34,35,39)(9,31,32,50,18,43,44,45)(14,36,38,61,37,60,62,64)(27,53,54,63,30,56,57,59)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=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, 321014);