#513204

order	512
nilpotency class	5
isoclinism	isoclinic class of D₄×D₃₂
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
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

359 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 359 conjugacy classes of subgroups — too many to draw. Only its 114 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)(9,10,15,21)(11,16,25,20)(12,26,30,59)(13,31,34,65)(14,35,37,40)(17,41,43,27)(18,44,47,33)(19,48,50,24)(22,52,29,63)(23,55,58,64)(28,61,62,71)(32,49,67,72)(36,45,54,70)(38,60,51,53)(39,68,46,57)(42,56,69,66);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,21,20)(12,22,30,29)(13,23,34,58)(14,24,37,48)(17,38,43,51)(18,39,47,46)(19,40,50,35)(26,52,59,63)(27,53,41,60)(28,54,62,36)(31,55,65,64)(32,56,67,66)(33,57,44,68)(42,49,69,72)(45,61,70,71);(9,12)(10,17)(11,22)(13,32)(14,36)(15,30)(16,38)(18,45)(19,49)(20,51)(21,43)(23,56)(24,28)(25,29)(26,31)(27,33)(34,67)(35,42)(37,54)(39,61)(40,69)(41,44)(46,71)(47,70)(48,62)(50,72)(52,55)(53,57)(58,66)(59,65)(60,68)(63,64);(3,8)(5,7)(9,11,15,25)(10,16,21,20)(12,22,30,29)(13,23,34,58)(14,24,37,48)(17,38,43,51)(18,39,47,46)(19,40,50,35)(26,52,59,63)(27,53,41,60)(28,54,62,36)(31,55,65,64)(32,56,67,66)(33,57,44,68)(42,49,69,72)(45,61,70,71)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,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=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420400);