#449115

order	512
nilpotency class	3
isoclinism	isoclinic class of #449115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15931
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=eg,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,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319687, #72319962, #72320069
maximal normal subgroup of	#72319687, #72319962, #72320069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356311);