#513039

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂³	C₂×D₁₆
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

388 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,6)(4,5)(7,8)(9,10,17,24)(11,19,25,46)(12,18,29,42)(13,21,34,53)(14,20,37,50)(15,38,40,67)(16,41,39,23)(22,54,55,32)(26,47,56,71)(27,61,57,72)(28,62,58,48)(30,43,63,68)(31,64,59,45)(33,65,35,52)(36,51,66,49)(44,70,69,60);(2,5)(6,8)(9,11,14,26,15,27,35,59,39,58,66,70,54,63,34,29,17,25,37,56,40,57,33,31,16,28,36,60,32,30,13,12)(10,18,21,43,22,44,51,62,41,64,65,72,67,71,50,46,24,42,53,68,55,69,49,48,23,45,52,61,38,47,20,19);(1,3,4,7)(2,6,5,8)(9,12,13,30,32,60,36,28,16,31,33,57,40,56,37,25,17,29,34,63,54,70,66,58,39,59,35,27,15,26,14,11)(10,19,20,47,38,61,52,45,23,48,49,69,55,68,53,42,24,46,50,71,67,72,65,64,41,62,51,44,22,43,21,18);(11,25)(12,29)(18,42)(19,46)(26,56)(27,57)(28,58)(30,63)(31,59)(43,68)(44,69)(45,64)(47,71)(48,62)(60,70)(61,72)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420235);