#419210

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15268
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1829 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)(4,21)(5,27,30,41)(6,13)(7,12)(8,14)(9,18)(10,25)(11,31,40,29)(16,44)(17,33,49,56)(19,35)(20,34)(22,32)(23,36,54,61)(24,38)(26,37)(28,43,48,39)(42,47)(45,60,51,62)(46,59)(50,53,57,55)(52,63,64,58);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,47,26,16,25,46)(5,28,29,49,30,48,31,17)(10,32,37,59,38,44,21,42)(11,39,27,56,40,43,41,33)(23,52,53,45,54,64,55,51)(36,60,50,63,61,62,57,58);(1,4)(2,10)(3,16)(5,23)(6,25)(7,24)(8,26)(9,32)(11,36)(12,21)(13,37)(14,38)(15,42)(17,45)(18,47)(19,46)(20,22)(27,57)(28,52)(29,55)(30,54)(31,53)(33,58)(34,59)(35,44)(39,62)(40,61)(41,50)(43,60)(48,64)(49,51)(56,63);(1,5)(2,11)(3,17)(4,23)(6,31)(7,29)(8,30)(9,33)(10,36)(12,41)(13,27)(14,40)(15,43)(16,45)(18,48)(19,28)(20,49)(21,50)(22,51)(24,55)(25,53)(26,54)(32,58)(34,39)(35,56)(37,57)(38,61)(42,60)(44,63)(46,52)(47,64)(59,62)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,e^d=ei,g^d=gi,h^d=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, 326406);