#501022

order	512
nilpotency class	4
isoclinism	isoclinic class of #501020
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

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

normal subgroups only

This group has 974 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)(3,13,16,8)(4,19,20,9)(5,11)(6,12)(7,10)(14,23,41,37)(15,30)(17,39,45,31)(18,36,28,32)(21,47,49,33)(22,48,50,34)(24,26)(25,29)(27,35)(38,44)(40,51)(42,56)(43,57)(46,55)(52,53,64,62)(54,58)(59,61)(60,63);(1,3,7,16)(2,8,10,13)(4,15,20,44)(5,23,27,37)(6,28,25,18)(9,30,19,38)(11,14,35,41)(12,36,29,32)(17,46,45,26)(21,51,49,59)(22,54,50,43)(24,31,55,39)(33,40,47,61)(34,58,48,57)(42,62,63,53)(52,56,64,60);(1,2)(3,14)(4,9)(5,24)(6,29)(7,10)(8,23)(11,26)(12,25)(13,37)(15,40)(16,41)(17,32)(18,31)(19,20)(21,52)(22,48)(27,55)(28,39)(30,51)(33,53)(34,50)(35,46)(36,45)(38,59)(42,57)(43,56)(44,61)(47,62)(49,64)(54,60)(58,63);(1,4)(2,9)(3,15)(5,21)(6,22)(7,20)(8,30)(10,19)(11,33)(12,34)(13,38)(14,40)(16,44)(17,42)(18,43)(23,51)(24,52)(25,50)(26,53)(27,49)(28,54)(29,48)(31,56)(32,57)(35,47)(36,58)(37,59)(39,60)(41,61)(45,63)(46,62)(55,64)

Polycyclic presentation:

2^9|a^2=f,b^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,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319877, #72343955, #72343961
maximal normal subgroup of	#72319877, #72343955, #72343961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408218);