#512960

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

352 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=e,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,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420156);