#494531

order	512
nilpotency class	4
isoclinism	isoclinic class of #33531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1879
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1281 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285775, #72285824, #72285937, …
maximal normal subgroup of	#72285775, #72285824, #72285937, …
maximal quotient of	#72321475, #72342569, #72342580

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401727);