#501026

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

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

normal subgroups only

This group has 971 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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, 408222);