#512964

order	512
nilpotency class	5
isoclinism	isoclinic class of #512827
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

350 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,21,23,50)(12,27,28,18)(14,33,31,64)(15,36,32,59)(17,41,38,24)(19,46,39,35)(20,48,40,30)(22,52,26,51)(25,57,55,71)(29,61,60,66)(34,49,65,70)(37,47,54,69)(42,53,44,56)(43,62,68,67)(45,63,58,72);(1,3,4,8)(2,5,6,7)(10,16)(17,38)(18,27)(19,39)(20,40)(21,50)(33,64)(36,59)(42,44)(43,68)(45,58)(47,69)(49,70)(51,52)(57,71)(61,66);(9,11)(10,17)(12,22)(13,23)(14,34)(15,37)(16,38)(18,42)(19,47)(20,49)(21,33)(24,35)(25,30)(26,28)(27,44)(29,62)(31,65)(32,54)(36,43)(39,69)(40,70)(41,46)(45,71)(48,55)(50,64)(51,66)(52,61)(53,63)(56,72)(57,58)(59,68)(60,67);(9,12,13,28)(10,18,16,27)(11,22,23,26)(14,29,31,60)(15,30,32,48)(17,42,38,44)(19,45,39,58)(20,36,40,59)(21,51,50,52)(24,53,41,56)(25,54,55,37)(33,66,64,61)(34,62,65,67)(35,63,46,72)(43,70,68,49)(47,71,69,57)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fg,g^a=gh,h^a=hi,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420160);