#513069

order	512
nilpotency class	5
isoclinism	isoclinic class of #13694
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	#13694	C₂
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

336 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 122 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)(11,17)(12,25)(13,31)(14,35)(15,37)(16,21)(18,27)(19,26)(20,32)(22,44)(23,48)(24,43)(28,58)(29,60)(30,51)(33,63)(34,62)(36,64)(38,46)(39,45)(40,66)(41,57)(42,54)(47,69)(49,52)(50,55)(53,61)(56,65)(59,72)(67,68)(70,71);(2,5)(6,8)(9,11,16,15)(10,17,21,37)(12,22,30,29)(13,23,34,33)(14,24,36,32)(18,38,41,65)(19,39,42,61)(20,35,43,64)(25,44,51,60)(26,45,54,53)(27,46,57,56)(28,47,59,55)(31,48,62,63)(40,49,67,70)(50,58,69,72)(52,68,71,66);(9,12,26,52,14,28,23,46,11,22,45,68,24,47,34,57,16,30,54,71,36,59,33,56,15,29,53,66,32,55,13,27)(10,18,31,50,20,40,61,60,37,65,63,72,64,70,42,51,21,41,62,69,43,67,39,44,17,38,48,58,35,49,19,25);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420265);