#511583

order	512
nilpotency class	4
isoclinism	isoclinic class of #511583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32707
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

601 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418779);