#507127

order	512
nilpotency class	4
isoclinism	isoclinic class of #501995
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28842
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=eg,d^2=fh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gh,h^c=hi

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, 414323);