#495755

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₄×B(4,𝔽₂)
prime radical	#33550	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1019 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,g^2=i,b^a=be,c^a=cf,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250212, #72283959, #72312758, …
maximal normal subgroup of	#72250212, #72283959, #72312758, …
maximal quotient of	#72257245, C₂×#495755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402951);