#508180

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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	#3061	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

814 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,26)(13,32)(14,20)(15,38)(16,21)(18,28)(19,35)(22,50)(23,53)(24,41)(25,43)(27,58)(29,47)(30,63)(31,60)(33,49)(34,48)(36,59)(37,42)(39,51)(40,55)(44,54)(45,61)(46,62)(52,71)(56,69)(57,68)(64,67)(65,70)(66,72);(2,5)(6,8)(9,11,16,25)(10,17,21,43)(12,27,31,57)(13,23,36,56)(14,37,15,24)(18,44,46,66)(19,40,48,70)(20,42,38,41)(22,30,52,29)(26,58,60,68)(28,54,62,72)(32,53,59,69)(33,64,61,51)(34,65,35,55)(39,49,67,45)(47,50,63,71);(3,7)(6,8)(9,12,16,31)(10,18,21,46)(11,22,25,52)(13,33,36,61)(14,30,15,29)(17,39,43,67)(19,47,48,63)(20,49,38,45)(23,54,56,72)(24,57,37,27)(26,59,60,32)(28,34,62,35)(40,68,70,58)(41,66,42,44)(50,69,71,53)(51,65,64,55);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415376);