#501024

order	512
nilpotency class	4
isoclinism	isoclinic class of #501024
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24354
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

984 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=f,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,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, 408220);