#513296

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, prime radical	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=ei,d^2=e,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, 420492);