#512924

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2262	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 458 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) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di,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, 420120);