#455833

order	512
nilpotency class	3
isoclinism	isoclinic class of #455832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16270
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1069 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)(3,14)(4,19)(5,11)(6,29)(7,12)(8,13)(9,17)(10,23)(15,35)(16,46)(18,33)(20,53)(21,36)(22,37)(24,52)(25,58)(26,41)(27,39)(28,40)(30,34)(31,55)(32,48)(38,57)(42,45)(43,49)(44,61)(47,60)(50,62)(51,56)(54,64)(59,63);(1,3)(2,9)(4,20,23,55)(5,16)(6,30)(7,17)(8,18)(10,35,19,42)(11,32)(12,33)(13,34)(14,29)(15,24,45,22)(21,54,56,47)(25,59,57,44)(26,50)(27,48)(28,49)(31,37,53,52)(36,61,51,63)(38,60,58,64)(39,43)(40,62)(41,46);(1,4,6,22,8,23,7,24)(2,10,12,37,13,19,29,52)(3,15,17,20,18,45,30,55)(5,25,26,56,28,57,27,21)(9,31,14,42,34,53,33,35)(11,38,39,51,40,58,41,36)(16,47,48,59,49,54,50,44)(32,61,46,64,62,63,43,60);(1,5,8,28)(2,11,13,40)(3,16,18,49)(4,21,23,56)(6,27,7,26)(9,32,34,62)(10,36,19,51)(12,41,29,39)(14,43,33,46)(15,44,45,59)(17,50,30,48)(20,54,55,47)(22,57,24,25)(31,60,53,64)(35,61,42,63)(37,58,52,38)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317961, #72318509, #72318589
maximal normal subgroup of	#72317961, #72318509, #72318589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363029);