#510899

order	512
nilpotency class	4
isoclinism	isoclinic class of #510898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32028
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

486 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,14)(4,20)(5,8)(6,13)(7,12)(9,16)(10,25)(11,22)(15,34)(17,40)(18,28)(19,43)(21,36)(23,24)(26,41)(27,37)(29,33)(30,42)(31,61)(32,57)(35,48)(38,58)(39,55)(44,46)(45,64)(47,60)(49,52)(50,62)(51,53)(54,56)(59,63);(1,3,6,16,8,19,22,18)(2,9,11,14,5,28,13,43)(4,21,24,53,27,54,25,49)(7,31,30,17,33,48,55,50)(10,36,37,52,23,56,20,51)(12,40,39,61,29,62,42,35)(15,32,45,59,47,58,46,26)(34,63,44,57,60,41,64,38);(1,4,22,25,8,27,6,24)(2,10,11,37,5,23,13,20)(3,15,16,45,19,47,18,46)(7,32,55,26,33,58,30,59)(9,34,43,64,28,60,14,44)(12,41,39,38,29,63,42,57)(17,49,48,21,50,53,31,54)(35,52,62,36,61,51,40,56);(1,5,8,2)(3,14,19,43)(4,23,27,10)(6,11,22,13)(7,29,33,12)(9,18,28,16)(15,44,47,64)(17,40,50,62)(20,24,37,25)(21,52,54,51)(26,57,59,38)(30,39,55,42)(31,61,48,35)(32,63,58,41)(34,46,60,45)(36,49,56,53)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418095);