#460350

order	512
nilpotency class	3
isoclinism	isoclinic class of #460350
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16362
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

1166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264914, #72264941, #72264996
maximal normal subgroup of	#72264914, #72264941, #72264996
maximal quotient of	#72265544

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367546);