#494606

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	#60387	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1866 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,f^b=fhi,d^c=di,e^c=eh,f^c=fi,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	#72281262, #72283959, #72312329, …
maximal normal subgroup of	#72281262, #72283959, #72312329, …
maximal quotient of	C₂×#494606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401802);