#494362

order	512
nilpotency class	4
isoclinism	isoclinic class of #33551
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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×#1867
prime radical	#3449	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1058 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282255, #72283959, #72312320, …
maximal normal subgroup of	#72282255, #72283959, #72312320, …
maximal quotient of	C₂×#494362

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401558);