#493543

order	512
nilpotency class	4
isoclinism	isoclinic class of #481288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#418
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

3135 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)(3,13)(4,18,20,9)(5,22)(6,28)(7,10)(8,15)(11,26)(12,25)(14,39,42,60)(16,40)(17,46)(19,49,24,57)(21,34,51,48)(23,38)(27,54)(29,36)(30,41,47,44)(31,45)(32,56)(33,52,37,58)(35,59)(43,62,53,64)(50,61,55,63);(1,3)(2,8)(4,14)(5,23)(6,29)(7,16)(9,30)(10,31)(11,36)(12,38)(13,22)(15,26)(17,25)(18,47)(19,50)(20,42)(21,53)(24,55)(27,56)(28,32)(33,62)(34,63)(35,46)(37,64)(39,49)(40,54)(41,52)(43,51)(44,58)(45,59)(48,61)(57,60);(1,4)(2,9)(3,14)(5,24)(6,21)(7,20)(8,30)(10,18)(11,37)(12,34)(13,39)(15,44)(16,42)(17,43)(19,27)(22,49)(23,55)(25,51)(26,58)(28,48)(29,53)(31,47)(32,61)(33,35)(36,64)(38,63)(40,60)(41,45)(46,62)(50,56)(52,59)(54,57);(2,10)(8,31)(9,18)(11,35)(12,28)(13,40)(22,54)(30,47)(32,38)(33,37)(34,48)(36,46)(39,60)(49,57)(61,63)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280034, #72280507, #72280993, …
maximal normal subgroup of	#72280034, #72280507, #72280993, …
maximal quotient of	#72285898, #72319010

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400739);