#423219

order	512
nilpotency class	3
isoclinism	isoclinic class of #423218
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15313
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60645	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1475 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)(5,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,31)(17,43)(19,33)(20,42)(22,34)(23,52)(24,37)(26,36)(27,56)(28,40)(29,39)(30,41)(32,46)(35,54)(38,47)(44,60)(45,49)(48,58)(50,55)(51,62)(53,61)(57,64)(59,63);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,42)(4,22,24,45,26,48,25,16)(5,27,28,47,30,46,29,17)(10,34,36,31,37,58,21,49)(11,38,39,56,41,43,40,32)(23,44,53,63,55,64,54,51)(35,60,50,62,61,57,52,59);(1,4)(2,10)(3,16,20,45)(5,23)(6,25)(7,24)(8,26)(9,31,33,49)(11,35)(12,21)(13,36)(14,37)(15,34,42,58)(17,44,47,64)(18,48,19,22)(27,51,46,63)(28,54)(29,53)(30,55)(32,57,56,60)(38,62,43,59)(39,52)(40,50)(41,61);(1,5,8,30)(2,11,14,41)(3,17,20,47)(4,23,26,55)(6,29,7,28)(9,32,33,56)(10,35,37,61)(12,40,13,39)(15,43,42,38)(16,44,45,64)(18,46,19,27)(21,50,36,52)(22,51,48,63)(24,54,25,53)(31,57,49,60)(34,59,58,62)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306351, #72306459, #72306467
maximal normal subgroup of	#72306351, #72306459, #72306467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330415);