#513301

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
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
prime radical	#2246	C₂²
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

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

normal subgroups only

This group has 329 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:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420497);