#494588

order	512
nilpotency class	4
isoclinism	isoclinic class of #33551
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1889
prime radical	#60398	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1886 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282473, #72283959, #72303182, …
maximal normal subgroup of	#72282473, #72283959, #72303182, …
maximal quotient of	C₂×#494588

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401784);