#359789

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13868
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1937 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317342, #72317348, #72317361
maximal normal subgroup of	#72317342, #72317348, #72317361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266985);