#501020

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

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

normal subgroups only

This group has 995 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)(3,13,15,8)(4,9)(5,11)(6,12)(7,10)(14,29,45,43)(16,44,47,30)(17,41,27,31)(18,32,50,60)(19,35)(20,33)(21,34)(22,53,54,38)(23,39)(24,37)(25,40)(26,36)(28,42)(46,57,49,59)(48,58,52,62)(51,61)(55,63)(56,64);(1,3)(2,8)(4,18)(5,22)(6,27)(7,15)(9,32)(10,13)(11,38)(12,41)(14,20)(16,25)(17,24)(19,50)(21,52)(23,49)(26,54)(28,48)(29,33)(30,40)(31,37)(34,62)(35,60)(36,53)(39,59)(42,58)(43,61)(44,64)(45,51)(46,55)(47,56)(57,63);(1,4)(2,9)(3,14)(5,23)(6,28)(7,19)(8,29)(10,35)(11,39)(12,42)(13,43)(15,45)(16,48)(17,49)(18,22)(20,25)(21,24)(26,55)(27,46)(30,58)(31,59)(32,38)(33,40)(34,37)(36,63)(41,57)(44,62)(47,52)(50,54)(51,56)(53,60)(61,64);(1,3)(2,8)(4,14)(5,16)(6,17)(7,15)(9,29)(10,13)(11,30)(12,31)(18,20)(19,45)(21,46)(22,25)(23,48)(24,27)(26,47)(28,49)(32,33)(34,57)(35,43)(36,44)(37,41)(38,40)(39,58)(42,59)(50,51)(52,55)(53,64)(54,56)(60,61)(62,63)

Polycyclic presentation:

2^9|a^2=f,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, #72319971, #72343953, …
maximal normal subgroup of	#72319877, #72319971, #72343953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408216);