#494327

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1864
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1043 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280077, #72280442, #72280652, …
maximal normal subgroup of	#72280077, #72280442, #72280652, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401523);