#502869

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60717	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

831 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=ei,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=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410065);