#496879

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404075);