#513148

order	512
nilpotency class	5
isoclinism	isoclinic class of #513086
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	#2215	C₂²
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

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

normal subgroups only

This group has 319 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=ce,f^c=fg,g^c=gh,h^c=hi

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, 420344);