#493547

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1786
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3129 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)(3,14)(4,19,22,37)(5,23)(6,13)(7,29)(8,11)(9,16)(10,21,35,28)(12,26)(15,43,46,63)(17,42)(18,41)(20,36,53,61)(24,33)(25,38)(27,55)(30,39)(31,58,59,51)(32,48)(34,56)(40,45,62,47)(44,49,64,60)(50,57,54,52);(1,3)(2,9)(4,15)(5,24)(6,17)(7,30)(8,18)(10,31)(11,32)(12,39)(13,33)(14,23)(16,26)(19,49)(20,51)(21,45)(22,46)(25,48)(27,56)(28,47)(29,34)(35,59)(36,43)(37,60)(38,42)(40,54)(41,55)(44,52)(50,62)(53,58)(57,64)(61,63);(1,4)(2,10)(3,15)(5,20)(6,28)(7,21)(8,22)(9,31)(11,35)(12,36)(13,37)(14,40)(16,44)(17,47)(18,46)(19,29)(23,54)(24,51)(25,57)(26,52)(27,53)(30,45)(32,59)(33,60)(34,49)(38,61)(39,43)(41,62)(42,63)(48,64)(50,55)(56,58);(2,11)(9,32)(10,35)(12,38)(13,29)(14,41)(19,37)(23,55)(31,59)(33,34)(36,61)(39,42)(40,62)(43,63)(49,60)(50,54)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280034, #72280192, #72280969, …
maximal normal subgroup of	#72280034, #72280192, #72280969, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400743);